Noether's theorem - Knowledge

10192: 9538: 10187:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T\right)&=\left({\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right){\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\left({\frac {d}{dt}}{\frac {\partial \varphi }{\partial \mathbf {q} }}\right){\dot {\mathbf {q} }}T+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\ddot {\mathbf {q} }}\,T\\&={\frac {\partial L}{\partial \mathbf {q} }}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\left({\frac {\partial ^{2}\varphi }{(\partial \mathbf {q} )^{2}}}{\dot {\mathbf {q} }}\right){\dot {\mathbf {q} }}T+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\ddot {\mathbf {q} }}\,T.\end{aligned}}} 9527: 10785: 11177: 8944: 10203: 10796: 22777: 11709: 9522:{\displaystyle {\begin{aligned}0={\frac {dI'}{d\varepsilon }}={}&L,{\dot {\mathbf {q} }},t_{2}]T-L,{\dot {\mathbf {q} }},t_{1}]T\\&{}+\int _{t_{1}}^{t_{2}}{\frac {\partial L}{\partial \mathbf {q} }}\left(-{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T+{\frac {\partial \varphi }{\partial \varepsilon }}\right)+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\left(-{\frac {\partial ^{2}\varphi }{(\partial \mathbf {q} )^{2}}}{\dot {\mathbf {q} }}^{2}T+{\frac {\partial ^{2}\varphi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}-{\frac {\partial \varphi }{\partial \mathbf {q} }}{\ddot {\mathbf {q} }}T\right)\,dt.\end{aligned}}} 10780:{\displaystyle {\begin{aligned}0={\frac {dI'}{d\varepsilon }}={}&L,{\dot {\mathbf {q} }},t_{2}]T-L,{\dot {\mathbf {q} }},t_{1}]T-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T\\&{}+\int _{t_{1}}^{t_{2}}{\frac {\partial L}{\partial \mathbf {q} }}{\frac {\partial \varphi }{\partial \varepsilon }}+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\varphi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}\,dt.\end{aligned}}} 11172:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}\right)=\left({\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right){\frac {\partial \varphi }{\partial \varepsilon }}+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\varphi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}={\frac {\partial L}{\partial \mathbf {q} }}{\frac {\partial \varphi }{\partial \varepsilon }}+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\varphi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}.} 11188: 14136: 11704:{\displaystyle {\begin{aligned}0={}&L,{\dot {\mathbf {q} }},t_{2}]T-L,{\dot {\mathbf {q} }},t_{1}]T-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}T\\&{}+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}.\end{aligned}}} 22763: 18696: 13703: 40: 21399: 8921: 1244: 2166: 7209: 18179: 14131:{\displaystyle {\begin{aligned}&\left\\={}&{\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right){\bar {\delta }}\varphi ^{A}+{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}\right).\end{aligned}}} 20926: 17028: 22598: 1938: 21084: 14362: 11872: 5396: 18691:{\displaystyle {\begin{aligned}q&=\int qd^{n}x\\&=\int \left\{\left({\frac {\partial }{\partial \varphi }}{\mathcal {L}}\right)\varepsilon Q+\left\partial _{\mu }(\varepsilon Q)\right\}d^{n}x\\&=\int \left\{\varepsilon Q+\partial _{\mu }\varepsilon \leftQ\right\}\,d^{n}x\\&\approx \int \varepsilon \partial _{\mu }\left\{f^{\mu }-\leftQ\right\}\,d^{n}x\end{aligned}}} 6999: 14565: 22126: 8504: 20028: 13570: 21814: 2938: 12065: 14725: 20679: 8493: 16694: 22348: 13400: 21394:{\displaystyle {\begin{aligned}{\vec {j}}&=\sum _{\alpha }\left({\frac {\partial }{\partial {\dot {\vec {x}}}_{\alpha }}}{\mathcal {L}}\right)\cdot {\vec {Q}}\left-{\vec {f}}\\&=\sum _{\alpha }\left(m_{\alpha }{\dot {\vec {x}}}_{\alpha }t-m_{\alpha }{\vec {x}}_{\alpha }\right)\\&={\vec {P}}t-M{\vec {x}}_{CM}\end{aligned}}} 12710: 4768: 14147: 11720: 8251: 2161:{\displaystyle \Delta S=\int \Delta L\approx \int {\frac {\partial L}{\partial {\dot {q}}}}\Delta {\dot {q}}\approx \int {\frac {\partial L}{\partial {\dot {q}}}}\left(\pm {\frac {\delta q}{\tau }}\right)\approx \ \pm {\frac {\partial L}{\partial {\dot {q}}}}\delta q=\pm {\frac {\partial L}{\partial {\dot {q}}}}\varphi .} 4381: 7204:{\displaystyle T_{\mu }{}^{\nu }=-\delta _{\mu }^{\nu }{\mathcal {L}}+\delta _{\mu }^{\sigma }\partial _{\sigma }\varphi {\frac {\partial {\mathcal {L}}}{\partial \varphi _{,\nu }}}=\left({\frac {\partial {\mathcal {L}}}{\partial \varphi _{,\nu }}}\right)\cdot \varphi _{,\mu }-\delta _{\mu }^{\nu }{\mathcal {L}}} 5258: 13692: 19286: 14373: 7791: 8916:{\displaystyle {\begin{aligned}I'&=\int _{t_{1}+\varepsilon T}^{t_{2}+\varepsilon T}L,{\dot {\mathbf {q} }}',t']\,dt'\\&=\int _{t_{1}+\varepsilon T}^{t_{2}+\varepsilon T}L,\varepsilon ],{\frac {\partial \varphi }{\partial \mathbf {q} }},\varepsilon ]{\dot {\mathbf {q} }},t']\,dt'\end{aligned}}} 4542: 5102:

is analogous to its linear momentum counterpart. It is assumed that the symmetry of the Lagrangian is rotational, i.e., that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations

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Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. Conversely, it allows researchers to

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as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity defined

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need not be involved, just evolution in time). For example, if the energy of a system is conserved, its energy is invariant at all times, which imposes a constraint on the system's motion and may help in solving for it. Aside from insights that such constants of motion give into the nature of a

17996: 3970: 19623: 19725: 13414: 21613: 11932: 12495: 14580: 20921:{\displaystyle {\begin{aligned}Q_{i}&=\sum _{\alpha }m_{\alpha }{\dot {x}}_{\alpha }^{i}-\sum _{\alpha <\beta }t\partial _{i}V_{\alpha \beta }\left({\vec {x}}_{\beta }-{\vec {x}}_{\alpha }\right)\\&=\sum _{\alpha }m_{\alpha }{\dot {x}}_{\alpha }^{i}.\end{aligned}}} 6833: 8052: 22703:

Invariance of an isolated system with respect to Lorentz boosts (i.e., that the laws of physics are the same with respect to all inertial reference frames) gives the center-of-mass theorem (which states that the center-of-mass of an isolated system moves at a constant

17023:{\displaystyle {\begin{aligned}Q\left&=\int _{N}\leftQ\,\mathrm {d} ^{n}x+\int _{\partial N}{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}Q\,\mathrm {d} s_{\mu }\\&\approx \int _{\partial N}f^{\mu }\,\mathrm {d} s_{\mu }.\end{aligned}}} 22593:{\displaystyle {\begin{aligned}j^{\mu }&=\leftQ-f^{\mu }\\&=\partial ^{\mu }\varphi \left(x^{\nu }\partial _{\nu }\varphi +\varphi \right)-x^{\mu }\left({\frac {1}{2}}\partial ^{\nu }\varphi \partial _{\nu }\varphi -\lambda \varphi ^{4}\right).\end{aligned}}} 6197: 8270: 13121: 19027: 20525: 12514: 4633: 14357:{\displaystyle \int _{\Omega }{\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}d^{4}x=0\,.} 15325: 11867:{\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}-L\right)T-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}} 4101: 5599: 15066: 17283: 1090:

of the system be conserved, as a consequence of its laws of motion. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. It is the laws of its motion that are symmetric.

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Although useful in its own right, the version of Noether's theorem just given is a special case of the general version derived in 1915. To give the flavor of the general theorem, a version of Noether's theorem for continuous fields in four-dimensional

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As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the

6380: 1050:. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on 7642: 5391:{\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot \mathbf {Q} =\mathbf {p} \cdot \left(\mathbf {n} \times \mathbf {r} \right)=\mathbf {n} \cdot \left(\mathbf {r} \times \mathbf {p} \right)=\mathbf {n} \cdot \mathbf {L} .} 5172: 4273: 7498: 4905: 12125: = 1, 2, 3). These four coordinates are the independent variables; and the values of the fields at each event are the dependent variables. Under an infinitesimal transformation, the variation in the coordinates is written 8071: 3048: 2250: 12796: 13581: 5878: 1933: 14560:{\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}=0\,.} 19072: 2941:

first term being due to stretching in temporal dimension of the "buffering" segment (that changes the size of the domain of integration), and the second is due to its "slanting" just as in the exemplar case. Together they add a summand

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which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the

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theorem: it relies on use of the equations of motion—the classical path. It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that

22281: 7657: 22121:{\displaystyle Q=\partial ^{\mu }\varphi \left(\partial _{\mu }\varphi +x^{\nu }\partial _{\mu }\partial _{\nu }\varphi +\partial _{\mu }\varphi \right)-4\lambda \varphi ^{3}\left(x^{\mu }\partial _{\mu }\varphi +\varphi \right).} 4422: 22665:

Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example:

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If the coordinates are changed, the boundary of the region of space–time over which the Lagrangian is being integrated also changes; the original boundary and its transformed version are denoted as Ω and Ω’, respectively.

16221: 6452: 21073: 6568:: that is, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point 12195: 1185:

The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the conserved quantity is called the

18184: 20023:{\displaystyle {\begin{aligned}j&=\sum _{i=1}^{3}{\frac {\partial L}{\partial {\dot {x}}_{i}}}Q-L\\&=m\sum _{i}{\dot {x}}_{i}^{2}-\left\\&={\frac {m}{2}}\sum _{i}{\dot {x}}_{i}^{2}+V(x).\end{aligned}}} 15639: 15135: 13565:{\displaystyle {\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\bar {\delta }}{\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}\left({\bar {\delta }}\varphi ^{A}\right)\,.} 6730: 20996: 7935: 6896: 3873: 21809:{\displaystyle {\begin{aligned}{\mathcal {S}}&=\int {\mathcal {L}}\leftd^{4}x\\&=\int \left({\frac {1}{2}}\partial ^{\mu }\varphi \partial _{\mu }\varphi -\lambda \varphi ^{4}\right)d^{4}x\end{aligned}}} 19390: 11924: 5049: 3767: 2933:{\displaystyle \Delta S\approx \pm \left(TL+\int {\frac {\partial L}{\partial {\dot {q}}_{r}}}\Delta {\dot {q}}_{r}\right)\approx \pm T\left(L-{\frac {\partial L}{\partial {\dot {q}}_{r}}}{\dot {q}}_{r}\right),} 17792: 16445: 15994: 15796: 5987: 4173: 3843: 2440: 21915: 12060:{\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\dot {\mathbf {q} }}-L\right)T-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}.} 1849: 21089: 19714: 14720:{\displaystyle {\begin{aligned}\delta x^{\mu }&=\varepsilon X^{\mu }\\\delta \varphi ^{A}&=\varepsilon \Psi ^{A}={\bar {\delta }}\varphi ^{A}+\varepsilon {\mathcal {L}}_{X}\varphi ^{A}\end{aligned}}} 19730: 17146: 12377: 17385: 5247: 2519: 21618: 20204: 13708: 11193: 10208: 9543: 8949: 8509: 5674: 17619: 17551: 6738: 7948: 8488:{\displaystyle {\dot {\mathbf {q} }}\rightarrow {\dot {\mathbf {q} }}'={\frac {d}{dt}}\varphi ,\varepsilon ]={\frac {\partial \varphi }{\partial \mathbf {q} }},\varepsilon ]{\dot {\mathbf {q} }}.} 150: 16529: 1111:

consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system. As an illustration, suppose that a physical theory is proposed which conserves a quantity

22353: 20684: 19077: 16699: 14585: 13395:{\displaystyle \left={\frac {\partial L}{\partial \varphi ^{A}}}{\bar {\delta }}\varphi ^{A}+{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}{\varphi ^{A}}_{,\sigma }\,.} 8076: 6102: 4427: 4278: 18168: 16634: 15522: 22337: 12801:

Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the

12705:{\displaystyle \int _{\Omega ^{\prime }}L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },\xi ^{\mu }\right)d^{4}\xi -\int _{\Omega }L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)d^{4}x=0} 1119:

through a continuous symmetry. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory.

4763:{\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\dot {\mathbf {q} }}-L\right)T_{r}-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot \mathbf {Q} _{r}} 3580: 2607: 18730: 14769: 20199: 7269: 6636: 20668: 1310: 6304: 22644: 15146: 1788: 3988: 5511: 3637:(gravitational stress–energy). The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant 14982: 17188: 12109:

ranges over the various components of the various tensor fields. These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates

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To avoid excessive complication of the formulas, this derivation assumed that the flow does not change as time passes. The same result can be obtained in the more general case.

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system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the suitable conservation laws.

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which, when multiplied by the charge on that species of particle, equals the electric current density due to that type of particle. This "gauge invariance" was first noted by

5998: 5932: 21474: 19374: 6319: 4376:{\displaystyle {\begin{aligned}t&\rightarrow t^{\prime }=t+\delta t\\\mathbf {q} &\rightarrow \mathbf {q} ^{\prime }=\mathbf {q} +\delta \mathbf {q} ~,\end{aligned}}} 16283: 16133: 16085: 15871: 15730: 15373: 5812: 5788: 3122: 1705: 8246:{\displaystyle {\begin{aligned}t&\rightarrow t'=t+\varepsilon T\\\mathbf {q} &\rightarrow \mathbf {q} '=\varphi ,\varepsilon ]=\varphi ,\varepsilon ]\end{aligned}}} 23757: 18719: 15690: 15668: 15442: 14831: 14800: 12103: 7549: 5121: 2722: 13687:{\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right)={\frac {\partial L}{\partial \varphi ^{A}}}} 7410: 4830: 21431: 20580: 3656:

In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering invariants. A major advance came in 1788 with the development of

3465: 3392: 3310: 1674: 1645: 12721: 3230: 2945: 21938: 19281:{\displaystyle {\begin{aligned}{\mathcal {S}}&=\int L\left\,dt\\&=\int \left({\frac {m}{2}}\sum _{i=1}^{3}{\dot {x}}_{i}^{2}-V(x(t))\right)\,dt.\end{aligned}}} 18109: 18020: ≈ 0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let 17180: 16244: 16153: 16105: 16053: 16021: 15847: 15823: 6663: 6593: 5820: 5744: 5724: 5633: 5462: 3415: 3336: 2745: 2245: 1857: 1728: 1547: 1527: 1397: 20071: 7312: 6472: 3634: 23586:

Vladimir Cuesta; Merced Montesinos; José David Vergara (2007). "Gauge invariance of the action principle for gauge systems with noncanonical symplectic structures".

5764: 5697: 3442: 3366: 3287: 3260: 2554: 2222: 2195: 1748: 1596: 1435: 7292: 6971: 6931: 6228: 3080: 1342: 21482: 20082: 7786:{\displaystyle j^{\nu }=i\left({\frac {\partial \psi }{\partial x^{\mu }}}\psi ^{*}-{\frac {\partial \psi ^{*}}{\partial x^{\mu }}}\psi \right)\eta ^{\nu \mu }~,} 20548: 7332: 6991: 6951: 6248: 6091: 4537:{\displaystyle {\begin{aligned}\delta t&=\sum _{r}\varepsilon _{r}T_{r}\\\delta \mathbf {q} &=\sum _{r}\varepsilon _{r}\mathbf {Q} _{r}~,\end{aligned}}} 3191: 2460: 1616: 1567: 1458: 1369: 1269: 1122:

There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the

22137: 3171: 1504: 12508:, which is defined as the integral of the Lagrangian density over the given region of spacetime. Expressed mathematically, this assumption may be written as 2379:{\displaystyle \left({\frac {\partial L}{\partial {\dot {q}}}}\varphi \right)(t_{0})=\left({\frac {\partial L}{\partial {\dot {q}}}}\varphi \right)(t_{1}),} 18091:

To see how the generalization is related to the version given above, assume that the action is the spacetime integral of a Lagrangian that only depends on

7826: 17626: 12323:{\displaystyle \varphi ^{A}\rightarrow \alpha ^{A}\left(\xi ^{\mu }\right)=\varphi ^{A}\left(x^{\mu }\right)+\delta \varphi ^{A}\left(x^{\mu }\right)\,.} 6316:

For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words,

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which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. For example,

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However, because the variations are defined at the same point as described above, the variation and the derivative can be done in reverse order; they

2612: 16161: 3629:, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of the 6389: 21004: 17991:{\displaystyle j_{12}^{\mu }=\left({\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}\right)(Q_{1}]-Q_{2}])-f_{12}^{\mu }.} 17556: 17488: 12131: 3965:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)={\frac {\partial L}{\partial \mathbf {q} }}~.} 1058:(developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a 19618:{\displaystyle Q={\frac {d}{dt}}\left=m\sum _{i}{\dot {x}}_{i}{\ddot {x}}_{i}-\sum _{i}{\frac {\partial V(x)}{\partial x_{i}}}{\dot {x}}_{i}} 8935: 15599: 15085: 6671: 23047:. In Proceedings of a Symposium on the Heritage of Emmy Noether, held on 2–4 December 1996, at the Bar-Ilan University, Israel, Appendix B. 6841: 20934: 11884: 7335: 4985: 3696: 246: 18024:

be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary.

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Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the

12490:{\displaystyle \alpha ^{A}\left(x^{\mu }\right)=\varphi ^{A}\left(x^{\mu }\right)+{\bar {\delta }}\varphi ^{A}\left(x^{\mu }\right)\,.} 4230:'s theory of general relativity By March 1918 she had most of the key ideas for the paper which would be published later in the year. 17043: 17331: 5211: 2465: 23825: 22696:(i.e., that the laws of physics are the same with respect to all angular orientations in space) gives the law of conservation of 6828:{\displaystyle {\mathcal {L}}\left(x^{\mu }\right)\mapsto {\mathcal {L}}\left(x^{\mu }-\varepsilon _{r}\delta _{r}^{\mu }\right)} 5641: 8047:{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}={\frac {\partial L}{\partial \mathbf {q} }}.} 23820: 23226:(15 October 1995). "A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes". 23730: 23470: 23389: 23331: 23196: 23111: 23076: 22946: 22878: 31: 6192:{\displaystyle j_{r}^{\nu }=\Lambda _{r}^{\nu }-{\frac {\partial {\mathcal {L}}}{\partial \varphi _{,\nu }}}\cdot \Psi _{r}} 30:

This article is about Emmy Noether's first theorem, which derives conserved quantities from symmetries. For other uses, see

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result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field

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And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a

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If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

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where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g.

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by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as

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problems, this field theory version is the most commonly used (or most often implemented) version of Noether's theorem.

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For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes

19022:{\displaystyle \partial _{\mu }\leftQ-2\left\partial _{\nu }Q+\partial _{\nu }\leftQ\right]-\,\dotsm \right]\approx 0.} 18117: 16544: 15481: 3982:, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that 983: 546: 22292: 20520:{\displaystyle {\begin{aligned}{\mathcal {S}}\left&=\int {\mathcal {L}}\leftdt\\&=\int \leftdt,\end{aligned}}} 69: 23444: 23289: 23206: 23179: 23144: 23028: 22913: 15467:

manifold and the target space is the set of values the fields can take at any given point. For example, if there are

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Leone, Raphaël (11 April 2018). "On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction".

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to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability

3859:)—the one actually taken by the system—is a path for which infinitesimal variations in that path cause no change in 3621:

was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of

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and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time

23671:"Symmetric energy–momentum tensor in Maxwell, Yang–Mills, and Proca theories obtained using only Noether's theorem" 23462: 23436: 23381: 22674:(in other words, that the laws of physics are the same at all locations in space) gives the law of conservation of 15320:{\displaystyle j^{\sigma }=\left-\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right)\Psi ^{A}\,.} 7220: 6598: 1078:

As an illustration, if a physical system behaves the same regardless of how it is oriented in space (that is, it's

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depends on the transformed coordinates ξ. To isolate the intrinsic changes, the field variation at a single point

4096:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{k}}}\right)={\frac {dp_{k}}{dt}}=0~,} 1274: 23805: 23170: 6263: 5594:{\displaystyle {\mathcal {S}}=\int {\mathcal {L}}\left(\varphi ,\partial _{\mu }\varphi ,x^{\mu }\right)\,d^{4}x} 823: 497: 477: 159: 22606: 20582:, consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words, 15061:{\displaystyle {\bar {\delta }}\varphi ^{A}=\varepsilon \Psi ^{A}-\varepsilon {\mathcal {L}}_{X}\varphi ^{A}\,.} 7388:| can be inferred from a set of measurements. Therefore, the system is invariant under transformations of the 1753: 16251: 7939:

is invariant under brief infinitesimal variations in the dependent variables. In other words, they satisfy the

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Several alternative methods for finding conserved quantities were developed in the 19th century, especially by

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Byers, Nina (1998). "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws".

17278:{\displaystyle J^{\mu }\,=\,{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}Q-f^{\mu },} 6313:

is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.

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Variational Formulation of Fluid and Geophysical Fluid Dynamics – Mechanics, Symmetries and Conservation Laws

22686: 16289: 15376: 13105:{\displaystyle \int _{\Omega }\left\{\left+{\frac {\partial }{\partial x^{\sigma }}}\left\right\}d^{4}x=0\,.} 7334:

obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see

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now comprises changes brought by every interval in play. Parts, where variation itself vanishes, i.e outside

1059: 605: 552: 438: 14966:{\displaystyle {\mathcal {L}}_{X}\varphi ^{A}={\frac {\partial \varphi ^{A}}{\partial x^{\mu }}}X^{\mu }\,.} 23815: 22806: 16258: 5887:. A system described by a given action might have multiple independent symmetries of this type, indexed by 4216: 264: 236: 14836: 7817:

Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables

6953:. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each 6561:{\displaystyle \varphi (x^{\mu })\mapsto \varphi \left(x^{\mu }-\varepsilon _{r}\delta _{r}^{\mu }\right)} 3499:

in the mathematical description of a system's evolution remains constant throughout its motion – it is an

347: 22713: 22650: 17460: 12336: 7940: 6063:{\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\varepsilon _{r}\partial _{\mu }\Lambda _{r}^{\mu }.} 5112: 4416:

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,

3864: 3672: 1123: 1016: 856: 469: 307: 279: 23396: 6375:{\displaystyle L\left({\boldsymbol {\varphi }},\partial _{\mu }{\boldsymbol {\varphi }},x^{\mu }\right)} 23810: 23373: 20073:(i.e. the principle of conservation of energy is a consequence of invariance under time translations). 5467: 3622: 727: 691: 473: 352: 241: 231: 15560: 15527: 5890: 3467:. That is the sense in which it is conserved: how much is "flowing" in, just as much is "flowing" out. 1379:

of the trajectory. In particular it would not change under a variation that applies the symmetry flow

23830: 21440: 19317: 15802: 7637:{\displaystyle L=\partial _{\nu }\psi \partial _{\mu }\psi ^{*}\eta ^{\nu \mu }+m^{2}\psi \psi ^{*}.} 7528: 5502: 5167:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\delta \theta \,\mathbf {n} \times \mathbf {r} .} 3661: 1032: 22971: 22646:(as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side). 21595:

Both examples 1 and 2 are over a 1-dimensional manifold (time). An example involving spacetime is a

16264: 16114: 16066: 15852: 15702: 15354: 7493:{\displaystyle \psi \rightarrow e^{i\theta }\psi \ ,\ \psi ^{*}\rightarrow e^{-i\theta }\psi ^{*}~,} 5793: 5769: 4900:{\displaystyle H={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\dot {\mathbf {q} }}-L.} 3085: 1679: 332: 21596: 16286: 15591: 6902: 5076: 4402: 4211: 3642: 3630: 3043:{\textstyle T\left(L-\sum _{r}\left(\partial L/\partial {\dot {q}}_{r}\right){\dot {q}}_{r}\right)} 626: 191: 18704: 15673: 15651: 15425: 14809: 14778: 12791:{\displaystyle {\varphi ^{A}}_{,\sigma }={\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}\,.} 12081: 2695: 1251:

The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate

6386: = 4, one for each dimension of space and time. An infinitesimal translation in space, 5873:{\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\varepsilon \partial _{\mu }\Lambda ^{\mu },} 3848: 2442:

is conserved, which is the conclusion of Noether's theorem. For instance if pure translations of

1928:{\displaystyle \Delta L\approx {\bigl (}\partial L/\partial {\dot {q}}{\bigr )}\Delta {\dot {q}}} 1345: 1079: 940: 732: 621: 23044: 21407: 20556: 7536: 3447: 3374: 3292: 1650: 1621: 1086:

is symmetric under continuous rotation: from this symmetry, Noether's theorem dictates that the

13115:

The difference in Lagrangians can be written to first-order in the infinitesimal variations as

4261: 4207: 4192:

from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of

3665: 3196: 1376: 1047: 976: 905: 866: 750: 686: 610: 21923: 20550:

Newtonian particles where the potential only depends pairwise upon the relative displacement.

18094: 17158: 16229: 16138: 16090: 16038: 16006: 15832: 15808: 6641: 6571: 5729: 5702: 5618: 5604:(the theorem can be further generalized to the case where the Lagrangian depends on up to the 5447: 3397: 3318: 2727: 2227: 1710: 1532: 1509: 1382: 23454: 23283: 22748: 22671: 20182: 20036: 16108: 15415: 7358: 7297: 6457: 5501:

would be representative of such a field, being a number defined at every place and time. The

1055: 950: 616: 392: 337: 298: 204: 23018: 21580:{\displaystyle {\frac {d{\vec {j}}}{dt}}=0\Rightarrow {\vec {P}}-M{\dot {\vec {x}}}_{CM}=0.} 20171:{\displaystyle \sum _{i=1}^{3}{\frac {\partial L}{\partial {\dot {x}}_{i}}}{\dot {x_{i}}}-L} 1598:. The only remaining parts are the "buffering" pieces. In these regions both the coordinate 23776: 23695: 23628: 23595: 23515: 23311: 23247: 22983: 22709: 5749: 5682: 3684: 3657: 3420: 3344: 3265: 3238: 2532: 2200: 2173: 1733: 1420: 1083: 1028: 955: 935: 861: 530: 454: 428: 342: 23404: 7277: 6956: 6916: 6213: 5413:

is arbitrary, i.e., if the system is insensitive to any rotation, then every component of

3056: 1572: 1318: 8: 22811: 22685:

translation (i.e. that the laws of physics are the same at all points in time) gives the

22276:{\displaystyle \partial _{\mu }\left=\partial _{\mu }\left(x^{\mu }{\mathcal {L}}\right)} 21604: 21600: 17152: 16664: 16024: 15695:

To get to the usual version of Noether's theorem, we need additional restrictions on the

15411: 15340: 14571: 4774: 4580: 4564: 4239: 4200: 3646: 3500: 1179: 1151: 1067: 1051: 1043: 1000: 930: 900: 890: 777: 631: 433: 289: 172: 167: 23780: 23699: 23632: 23599: 23519: 23315: 23251: 23045:"E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws" 22987: 23766: 23741: 23685: 23649: 23549: 23531: 23505: 23337: 23271: 23237: 22821: 22768: 22733: 20533: 16681: 16000: 12802: 8058: 7532: 7317: 6976: 6936: 6233: 6076: 5884: 5814:

is left invariant, but it will also be true if the Lagrangian changes by a divergence,

5059: 5055: 4405:

of the action, i.e. transformations leaving the action unchanged; labelled by an index

3626: 3586: 3478: 3176: 2445: 1601: 1552: 1443: 1354: 1254: 1163: 1036: 1008: 895: 798: 782: 722: 717: 712: 676: 557: 481: 387: 382: 186: 181: 23755:

Sardanashvily (2009). "Gauge conservation laws in a general setting. Superpotential".

23662: 23093: 4249:

The essence of Noether's theorem is generalizing the notion of ignorable coordinates.

3130: 1463: 23840: 23726: 23535: 23466: 23440: 23385: 23341: 23327: 23263: 23228: 23202: 23175: 23165: 23140: 23132: 23107: 23072: 23058: 23024: 22999: 22952: 22942: 22919: 22909: 22884: 22874: 22762: 22654: 16060: 7393: 7354: 6094: 5505:

can be applied to such fields, but the action is now an integral over space and time

4915:

Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate

4256:

defined above is invariant under small perturbations (warpings) of the time variable

3610: 3486: 3339: 1175: 1155: 969: 803: 581: 464: 417: 274: 269: 23020:

Angular Momentum: an illustrated guide to rotational symmetries for physical systems

18000:

This shows we can extend Noether's theorem to larger Lie algebras in a natural way.

4203:

under such transformations. This is the seed idea generalized in Noether's theorem.

3606: 23784: 23636: 23603: 23523: 23319: 23275: 23255: 23099: 23064: 22991: 22796: 22791: 22737: 22729: 22697: 20076:

More generally, if the Lagrangian does not depend explicitly on time, the quantity

19300: 19296: 19059: 17305: 15696: 15645: 15445: 12505: 8264:

is a real constant (which could be zero) indicating how much the flow shifts time.

6254: 5418: 5088: 3676: 3492: 3482: 3368:, that is built in a way analogous to the prior definition of a conserved quantity. 1372: 1349: 1171: 1096: 1087: 1012: 1004: 813: 707: 681: 542: 459: 423: 23585: 17448:{\textstyle \left\langle \int d^{4}x~\partial \cdot {\textbf {J}}\right\rangle =0} 22816: 22717: 22675: 15456: 15380: 7342: 6475: 6310: 5442: 5434:

is now given. Since field theory problems are more common in modern physics than

4227: 4222:

Emmy Noether's work on the invariance theorem began in 1915 when she was helping

2682:{\textstyle \sum _{r}\left(\partial L/\partial {\dot {q}}_{r}\right)\varphi _{r}} 945: 818: 772: 767: 654: 567: 512: 22700:(which states that the total angular momentum of an isolated system is constant) 1529:. The middle part does not change the action either, because its transformation 1234:

linearly independent combinations of the Lagrangian expressions are divergences.

23607: 23424: 23378:

The Noether theorems: Invariance and conservation laws in the twentieth century

22801: 22782: 22725: 22678:(which states that the total linear momentum of an isolated system is constant) 19292: 18036:

apply to the boundary, by the way), the derivation distribution q generated by

17485:. Then, is also a symmetry derivation. Let us see this explicitly. Let us say 14772: 7801: 7540: 7377: 5431: 4574: 3602: 2724:, they cause the "buffering" segments to contribute the two following terms to 828: 636: 408: 47:'s article "Invariante Variationsprobleme" (1918), where she proved her theorem 23788: 23527: 23323: 23103: 23068: 22995: 23799: 23703: 23567: 23358: 23350: 23259: 23003: 22956: 22888: 18029: 17390:

The quantum analogs of Noether's theorem involving expectation values (e.g.,

16032: 13406: 4557: 1138:

All fine technical points aside, Noether's theorem can be stated informally:

1024: 808: 572: 327: 284: 23616: 22923: 16216:{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\approx 0} 5883:

since the integral of a divergence becomes a boundary term according to the

3664:. In this approach, the state of the system can be described by any type of 23488: 23380:. Sources and Studies in the History of Mathematics and Physical Sciences. 23267: 22721: 21920:

The second term on the right hand side is due to the "conformal weight" of

15475: 15392: 7797: 6447:{\displaystyle x^{\mu }\mapsto x^{\mu }+\varepsilon _{r}\delta _{r}^{\mu }} 4620: 3618: 2224:, should cancel each other in order to make the total change in the action 1020: 562: 312: 44: 39: 23740:

Quigg, Chris (9 July 2019). "Colloquium: A Century of Noether's Theorem".

22868: 21068:{\displaystyle {\vec {f}}=\sum _{\alpha }m_{\alpha }{\vec {x}}_{\alpha }.} 1146:

A more sophisticated version of the theorem involving fields states that:

23223: 22903: 15471: 14976:

These equations imply that the field variation taken at one point equals

11877:

is a constant of the motion, i.e., it is a conserved quantity. Since φ =

7647:

In this case, Noether's theorem states that the conserved (∂ ⋅

6203: 5790:

is left invariant. This will certainly be true if the Lagrangian density

4223: 2462:

by a constant are the symmetry, then the conserved quantity becomes just

1243: 1206: 1063: 925: 23554: 23510: 12190:{\displaystyle x^{\mu }\rightarrow \xi ^{\mu }=x^{\mu }+\delta x^{\mu }} 5252:

Then Noether's theorem states that the following quantity is conserved,

1166:

of the form that a physical law takes with respect to a one-dimensional

1162:

The word "symmetry" in the above statement refers more precisely to the

23690: 17301: 15826: 15554: 6901:

and thus Noether's theorem corresponds to the conservation law for the

5609: 3638: 3508: 2247:

be zero, as would be expected if the trajectory is a solution. That is

1199: 1127: 671: 595: 322: 317: 221: 23640: 23242: 22689:(which states that the total energy of an isolated system is constant) 18724:

More generally, if the Lagrangian depends on higher derivatives, then

15634:{\displaystyle {\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb {R} ,} 15130:{\displaystyle {\frac {\partial }{\partial x^{\sigma }}}j^{\sigma }=0} 6725:{\displaystyle \Psi _{r}=-\delta _{r}^{\mu }\partial _{\mu }\varphi .} 23617:"Symmetries and conservation laws: Consequences of Noether's theorem" 23563: 20991:{\textstyle {\frac {d}{dt}}\sum _{\alpha }m_{\alpha }x_{\alpha }^{i}} 20033:

The right hand side is the energy, and Noether's theorem states that

17456: 17312:

is noncompact, the currents fall off sufficiently fast at infinity).

16670: 16644: 15464: 7930:{\displaystyle I=\int _{t_{1}}^{t_{2}}L,{\dot {\mathbf {q} }},t]\,dt} 6891:{\displaystyle \Lambda _{r}^{\mu }=-\delta _{r}^{\mu }{\mathcal {L}}} 5435: 4397:

are both small, but variable. For generality, assume there are (say)

3645:. Another important conserved quantity, discovered in studies of the 3614: 1167: 600: 590: 23194: 15692:; this is for physical reasons, and is unimportant for this proof.) 11919:{\displaystyle {\frac {\partial \varphi }{\partial \mathbf {q} }}=1} 5044:{\displaystyle p_{k}={\frac {\partial L}{\partial {\dot {q_{k}}}}}.} 3762:{\displaystyle I=\int L(\mathbf {q} ,{\dot {\mathbf {q} }},t)\,dt~,} 1115:. A researcher can calculate the types of Lagrangians that conserve 1031:

function, from which the system's behavior can be determined by the

27:

Statement relating differentiable symmetries to conserved quantities

23746: 23654: 22693: 21823:, consider the generator of a spacetime rescaling. In other words, 17787:{\displaystyle =Q_{1}]-Q_{2}]\approx \partial _{\mu }f_{12}^{\mu }} 17321: 17297: 17293: 16685: 16667: 16652: 16640: 16440:{\displaystyle Q\left\approx \int _{\partial N}f^{\mu }\,ds_{\mu }} 16247: 15989:{\displaystyle {\mathcal {S}}\,=\,\int _{M}{\mathcal {L}}\,d^{n}x.} 15791:{\displaystyle {\mathcal {L}}(\varphi ,\partial _{\mu }\varphi ,x)} 15733: 15553:. If the field is a real vector field, then the target manifold is 5982:{\displaystyle \varphi \mapsto \varphi +\varepsilon _{r}\Psi _{r},} 4969: 4968: = 1; the conserved quantity is the corresponding linear 4168:{\displaystyle p_{k}={\frac {\partial L}{\partial {\dot {q}}_{k}}}} 3838:{\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}~.} 3598: 2435:{\displaystyle \left(\partial L/\partial {\dot {q}}\right)\varphi } 1100: 666: 413: 370: 59: 23771: 21910:{\displaystyle Q=x^{\mu }\partial _{\mu }\varphi (x)+\varphi (x).} 12200:

whereas the transformation of the field variables is expressed as

4818: = 0; the corresponding conserved quantity is the total 1844:{\displaystyle {\dot {q}}\rightarrow {\dot {q}}\pm \delta q/\tau } 1730:

in the coordinate is negligible by comparison since the time span

1247:

Plot illustrating Noether's theorem for a coordinate-wise symmetry

22744: 19709:{\displaystyle L={\frac {m}{2}}\sum _{i}{\dot {x}}_{i}^{2}-V(x).} 15079: = 0 and changing the sign yields the conservation law 4608:; for rotations, it is an expression linear in the components of 1238: 22847:

Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen

22716:, yields further conservation laws, such as the conservation of 20188: 5934:

so the most general symmetry transformation would be written as

3585:

Such quantities are said to be conserved; they are often called

23668: 23614: 23405:"Understanding Noether's Theorem by Visualizing the Lagrangian" 17141:{\displaystyle \partial _{\mu }\left-f^{\mu }\right]\approx 0.} 4819: 4605: 1104: 23459:

Noether's Theorems. Applications in Mechanics and Field Theory

22842: 18088:

in general). This is the generalization of Noether's theorem.

14367:

Since this holds for any region Ω, the integrand must be zero

5464:

defined over all space and time; for example, the temperature

1170:

of transformations satisfying certain technical criteria. The

19042:

Looking at the specific case of a Newtonian particle of mass

17380:{\displaystyle \int _{\partial N}J^{\mu }ds_{\mu }\approx 0.} 15449: 5242:{\displaystyle \mathbf {Q} =\mathbf {n} \times \mathbf {r} .} 4613: 3417:

is interpreted as vanishing of the total flux of the current

2514:{\displaystyle \left(\partial L/\partial {\dot {q}}\right)=p} 1195: 23758:

International Journal of Geometric Methods in Modern Physics

22970:

De Azcárraga, J.a.; Lukierski, J.; Vindel, P. (1986-07-01).

22653:

analog of this equation, one runs into a problem because of

16246:

satisfies the given boundary conditions, is the subspace of

3863:, at least up to first order. This principle results in the 2609:, their effects add up by linearity to a conserved quantity 1790:. So the regions contribute mostly through their "slanting" 22969: 22941:(5th ed.). Boston, MA: Brooks/Cole, Cengage Learning. 22682: 22286:(where we have performed a change of dummy indices) so set 5669:{\displaystyle \varphi \mapsto \varphi +\varepsilon \Psi ,} 4178:

is conserved throughout the motion (on the physical path).

3512: 7314:

at an intermediate step to avoid conflict). (However, the

6073:

For such systems, Noether's theorem states that there are

22670:

Invariance of an isolated system with respect to spatial

18032:. Then, because of the variational principle (which does 17614:{\displaystyle Q_{2}\approx \partial _{\mu }f_{2}^{\mu }} 17546:{\displaystyle Q_{1}\approx \partial _{\mu }f_{1}^{\mu }} 1375:

on this trajectory, i.e. does not change under any local

12078:

Noether's theorem may also be derived for tensor fields

4199:; the Lagrangian is invariant, and is said to exhibit a 1750:

of the buffering is small (taken to the limit of 0), so

13697:

the difference in Lagrangians can be written neatly as

11182:

Substituting this into the previous equation, one gets

10197:

Substituting this into the previous equation, one gets

1217:

If an integral I is invariant under a continuous group

21590: 20937: 17396: 8260:

is a real variable indicating the amount of flow, and

5401:

In other words, the component of the angular momentum

5062:, these two conservation laws can be expressed either 2948: 2615: 1575: 23496:

Emmy Noether (1971). "Invariant Variation Problems".

22609: 22351: 22295: 22140: 21949: 21926: 21832: 21616: 21485: 21443: 21410: 21087: 21007: 20682: 20591: 20559: 20536: 20202: 20085: 20039: 19728: 19637: 19393: 19320: 19075: 18733: 18707: 18182: 18120: 18097: 17821: 17629: 17559: 17491: 17334: 17191: 17161: 17046: 16697: 16547: 16463: 16305: 16267: 16232: 16164: 16141: 16117: 16093: 16069: 16041: 16009: 15881: 15855: 15835: 15811: 15749: 15705: 15676: 15654: 15602: 15563: 15530: 15484: 15428: 15357: 15149: 15088: 15071:

Differentiating the above divergence with respect to

14985: 14887: 14839: 14812: 14781: 14736: 14583: 14376: 14150: 13706: 13584: 13417: 13124: 12814: 12724: 12517: 12380: 12339: 12209: 12134: 12084: 11935: 11887: 11723: 11191: 10799: 10206: 9541: 8947: 8507: 8273: 8074: 7951: 7829: 7660: 7552: 7413: 7320: 7300: 7280: 7223: 7002: 6979: 6959: 6939: 6919: 6844: 6741: 6674: 6644: 6601: 6574: 6484: 6460: 6392: 6322: 6266: 6236: 6216: 6105: 6079: 6001: 5943: 5893: 5823: 5796: 5772: 5752: 5732: 5705: 5685: 5644: 5621: 5514: 5470: 5450: 5261: 5214: 5124: 4988: 4833: 4636: 4425: 4276: 4115: 3991: 3876: 3789: 3699: 3524: 3450: 3423: 3400: 3377: 3347: 3321: 3295: 3268: 3241: 3199: 3179: 3133: 3088: 3059: 2754: 2730: 2698: 2562: 2535: 2468: 2448: 2392: 2253: 2230: 2203: 2176: 1941: 1860: 1796: 1756: 1736: 1713: 1682: 1653: 1624: 1604: 1555: 1535: 1512: 1466: 1446: 1423: 1385: 1357: 1321: 1277: 1257: 1073: 1035:. This theorem only applies to continuous and smooth 72: 23429:

Applications of Lie groups to differential equations

22758: 22720:

from the invariance with respect to a change in the

19037: 16651:

generates an off-shell symmetry. If this only holds

16524:{\displaystyle Q\approx \partial _{\mu }f^{\mu }(x)} 15391:. (More generally, we can have smooth sections of a 3617:

experiments, and refined by subsequent researchers.

1190:, while the flow carrying that charge is called the 1133: 23372: 22743:The Noether charge is also used in calculating the 17308:called the Noether charge (provided, of course, if 15330: 6735:The Lagrangian density transforms in the same way, 5766:to generate a physical symmetry is that the action 3635:

Landau–Lifshitz stress–energy–momentum pseudotensor

22638: 22592: 22331: 22275: 22120: 21932: 21909: 21808: 21579: 21468: 21425: 21393: 21067: 20990: 20920: 20662: 20574: 20542: 20519: 20170: 20065: 20022: 19708: 19617: 19368: 19280: 19021: 18713: 18690: 18162: 18103: 17990: 17786: 17613: 17545: 17447: 17379: 17277: 17174: 17140: 17022: 16628: 16523: 16439: 16277: 16238: 16215: 16147: 16127: 16099: 16079: 16047: 16015: 15988: 15865: 15841: 15817: 15790: 15724: 15684: 15662: 15633: 15578: 15545: 15516: 15436: 15367: 15319: 15129: 15060: 14965: 14870: 14825: 14794: 14763: 14719: 14559: 14356: 14130: 13686: 13564: 13394: 13104: 12790: 12704: 12489: 12355: 12322: 12189: 12097: 12059: 11918: 11866: 11703: 11171: 10779: 10186: 9521: 8915: 8487: 8245: 8046: 7929: 7785: 7636: 7492: 7326: 7306: 7286: 7263: 7203: 6985: 6965: 6945: 6925: 6890: 6827: 6724: 6657: 6630: 6587: 6560: 6466: 6446: 6374: 6298: 6242: 6222: 6191: 6085: 6062: 5981: 5926: 5872: 5806: 5782: 5758: 5738: 5718: 5691: 5668: 5627: 5593: 5493: 5456: 5390: 5241: 5166: 5043: 4899: 4762: 4536: 4375: 4244: 4167: 4095: 3964: 3837: 3761: 3574: 3459: 3436: 3409: 3386: 3360: 3330: 3304: 3281: 3254: 3224: 3185: 3165: 3116: 3074: 3042: 2932: 2739: 2716: 2681: 2601: 2548: 2513: 2454: 2434: 2378: 2239: 2216: 2189: 2160: 1927: 1843: 1782: 1742: 1722: 1699: 1668: 1639: 1610: 1590: 1561: 1541: 1521: 1498: 1452: 1429: 1391: 1363: 1344:(bold on the diagram) that satisfies the system's 1336: 1304: 1263: 144: 23050: 22972:"Superfields and canonical methods in superspace" 22728:field of the charged particle and the associated 22692:Invariance of an isolated system with respect to 22681:Invariance of an isolated system with respect to 21476:is the center of mass. Noether's theorem states: 18163:{\displaystyle Q\approx \partial _{\mu }f^{\mu }} 17466: 16629:{\displaystyle {\mathcal {L}}(x)={\mathcal {L}}.} 15517:{\displaystyle \varphi _{1},\ldots ,\varphi _{m}} 14574:transformations, the perturbation can be written 14141:Thus, the change in the action can be written as 6382:is constant in its third argument. In that case, 5699:is in general a function that may depend on both 3597:The earliest constants of motion discovered were 2170:These last terms, evaluated around the endpoints 23797: 23160: 23158: 23156: 22332:{\displaystyle f^{\mu }=x^{\mu }{\mathcal {L}}.} 15444:, representing time and the target space is the 10790:Again using the Euler–Lagrange equations we get 7503:a complex rotation. In the limit when the phase 4924:; so it is invariant (symmetric) under changes 3776:signifies the rate of change of the coordinates 3671:; the laws of motion need not be expressed in a 1549:is a symmetry and thus preserves the Lagrangian 1070:need not have a corresponding conservation law. 1023:in 1918. The action of a physical system is the 145:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)} 23195:Michael E. Peskin; Daniel V. Schroeder (1995). 22937:Thornton, Stephen T.; Marion, Jerry B. (2004). 9532:Notice that the Euler–Lagrange equations imply 4560:parameter coefficients corresponding to each: 4409: = 1, 2, 3, ..., 3675:, as was customary in Newtonian mechanics. The 3575:{\displaystyle {\frac {dX}{dt}}={\dot {X}}=0~.} 2602:{\displaystyle q_{r}\mapsto q_{r}+\varphi _{r}} 1154:generated by local actions there corresponds a 23720: 23351:"Noether's Theorem: Symmetry and Conservation" 23305: 23139:(4th ed.). New York: Dover Publications. 23127: 23125: 23123: 23016: 22936: 18003: 17296:. The continuity equation tells us that if we 14764:{\displaystyle {\mathcal {L}}_{X}\varphi ^{A}} 12121: = 0) and three spatial dimensions ( 4226:and David Hilbert with their work related to 3371:Now, the zero contribution of the "buffering" 1437:to transition between the segments gradually. 1239:Brief illustration and overview of the concept 23754: 23453: 23174:(2nd ed.). Reading, MA: Addison-Wesley. 23153: 20189:Example 2: Conservation of center of momentum 16659:generates an on-shell symmetry. Then, we say 12073: 7345:, by contrast, can be derived by considering 7264:{\displaystyle T_{\mu }{}^{\nu }{}_{,\nu }=0} 6631:{\displaystyle x^{\mu }-\varepsilon X^{\mu }} 5608:derivative, and can also be formulated using 3605:, which were proposed in the 17th century by 1905: 1872: 1209:'s statement of Noether's theorem for action 977: 23615:Hanca, J.; Tulejab, S.; Hancova, M. (2004). 23495: 23486: 23306:Badin, Gualtiero; Crisciani, Fulvio (2018). 20663:{\displaystyle Q_{i}\left=t\delta _{i}^{j}.} 19050:, moving under the influence of a potential 11926:and so the conserved quantity simplifies to 7543:particles, which has the Lagrangian density 5079:(this will be derived in the next section). 1305:{\displaystyle \varphi :q\mapsto q+\delta q} 23120: 23098:. Cham: Springer International Publishing. 23095:Emmy Noether – Mathematician Extraordinaire 22939:Classical dynamics of particles and systems 22870:Classical Dynamics: A Contemporary Approach 22867:José, Jorge V.; Saletan, Eugene J. (1998). 22866: 19380:has an explicit dependence on time, whilst 7812: 7527:*, respectively. A specific example is the 6299:{\displaystyle \partial _{\nu }j^{\nu }=0,} 23669:Merced Montesinos; Ernesto Flores (2006). 22873:. Cambridge : Cambridge University Press. 22639:{\displaystyle \partial _{\mu }j^{\mu }=0} 20193:Still considering 1-dimensional time, let 5615:A continuous transformation of the fields 4233: 3633:(non-gravitational stress–energy) and the 3503:. Mathematically, the rate of change of 1783:{\displaystyle \delta q/\tau \gg \delta q} 1126:. Generalizations of Noether's theorem to 984: 970: 23770: 23745: 23689: 23653: 23553: 23509: 23241: 23164: 22908:. Cambridge: Cambridge University Press. 22901: 19295:of the particle, while the second is its 19264: 19155: 19004: 18670: 18547: 17471:Suppose we have two symmetry derivations 17206: 17202: 16997: 16948: 16864: 16727: 16423: 16331: 15969: 15902: 15898: 15678: 15656: 15624: 15566: 15533: 15430: 15313: 15237: 15054: 14959: 14553: 14350: 13575:Using the Euler–Lagrange field equations 13558: 13388: 13098: 12784: 12483: 12333:By this definition, the field variations 12316: 10763: 10173: 9911: 9505: 8897: 8663: 8065:, which acts on the variables as follows 7920: 5577: 5147: 3746: 1015:. This is the first of two theorems (see 105: 23504:(3). Translated by Mort Tavel: 186–207. 23498:Transport Theory and Statistical Physics 23221: 22902:Hand, Louis N.; Finch, Janet D. (1998). 18111:and its first derivatives. Also, assume 16003:, i.e., a specification of the value of 5193:parameter and the Cartesian coordinates 4802:, without any change in the coordinates 4210:. For example, he developed a theory of 2524:More general cases follow the same idea: 1242: 38: 23348: 23198:An Introduction to Quantum Field Theory 23137:The Variational Principles of Mechanics 23131: 22840: 22712:, the analog to Noether's theorem, the 21599:of a massless real scalar field with a 18072:not on the boundary (but remember that 8926:which may be regarded as a function of 6638:"behind" it which would be mapped onto 6350: 6332: 6257:is expressed in a four-dimensional way 5424: 3124:are considered, the argument replaces 508:Differentiating under the integral sign 14: 23798: 23188: 19291:The first term in the brackets is the 15829:and the position. In other words, for 3053:Finally, when instead of a trajectory 23739: 23647: 23547: 23423: 23402: 22849:. Mathematisch-Physikalische Klasse. 5441:Let there be a set of differentiable 5177:Since time is not being transformed, 3975:Thus, if one of the coordinates, say 3472: 23562: 23091: 23056: 22862: 22860: 14871:{\displaystyle {X^{\mu }}_{,\nu }=0} 2692:When there are time transformations 21591:Example 3: Conformal transformation 18084:, not a derivation parametrized by 17429: 15140:where the conserved current equals 14570:For any combination of the various 12356:{\displaystyle \delta \varphi ^{A}} 7511:, it may be taken as the parameter 7368:) of finding a particle at a point 7353:rather than in the derivatives. In 4252:One can assume that the Lagrangian 24: 23725:. Johns Hopkins University Press. 23721:Neuenschwander, Dwight E. (2010). 23299: 22611: 22550: 22537: 22485: 22457: 22411: 22391: 22384: 22380: 22321: 22263: 22237: 22190: 22177: 22142: 22092: 22040: 22024: 22014: 21988: 21970: 21958: 21868: 21756: 21743: 21675: 21649: 21623: 21166: 21132: 21128: 20784: 20702: 20246: 20209: 20118: 20110: 19775: 19767: 19580: 19563: 19082: 18974: 18954: 18944: 18937: 18933: 18912: 18887: 18876: 18856: 18846: 18839: 18835: 18800: 18780: 18773: 18769: 18735: 18643: 18623: 18616: 18612: 18578: 18520: 18498: 18489: 18485: 18466: 18454: 18377: 18366: 18346: 18339: 18335: 18300: 18289: 18285: 18239: 18204: 18141: 18129: 18080:) is a shorthand for a derivation 17878: 17858: 17851: 17847: 17760: 17745: 17703: 17664: 17587: 17575: 17519: 17507: 17421: 17340: 17229: 17222: 17215: 17210: 17084: 17077: 17070: 17065: 17048: 16999: 16979: 16950: 16918: 16911: 16904: 16899: 16888: 16867: 16829: 16822: 16815: 16810: 16798: 16785: 16778: 16773: 16730: 16722: 16593: 16569: 16550: 16493: 16472: 16408: 16405: 16396: 16360: 16334: 16326: 16270: 16173: 16120: 16072: 15939: 15915: 15884: 15858: 15767: 15752: 15708: 15615: 15605: 15360: 15304: 15271: 15263: 15210: 15179: 15171: 15095: 15091: 15033: 15015: 14933: 14918: 14891: 14740: 14692: 14649: 14415: 14407: 14383: 14379: 14199: 14191: 14167: 14163: 14156: 14066: 14058: 14034: 14030: 13963: 13955: 13895: 13887: 13864: 13860: 13668: 13660: 13622: 13614: 13591: 13587: 13510: 13506: 13485: 13470: 13326: 13318: 13274: 13266: 12974: 12970: 12820: 12768: 12753: 12617: 12528: 12524: 12045: 12037: 12014: 12006: 11952: 11944: 11899: 11891: 11855: 11847: 11824: 11816: 11771: 11763: 11740: 11732: 11669: 11661: 11638: 11630: 11599: 11591: 11568: 11560: 11501: 11493: 11470: 11462: 11412: 11404: 11381: 11373: 11141: 11135: 11121: 11097: 11089: 11074: 11066: 11052: 11044: 11013: 11007: 10993: 10969: 10961: 10946: 10938: 10910: 10902: 10862: 10854: 10831: 10823: 10738: 10732: 10718: 10694: 10686: 10671: 10663: 10649: 10641: 10553: 10545: 10522: 10514: 10464: 10456: 10433: 10425: 10148: 10140: 10117: 10109: 10046: 10029: 10000: 9992: 9958: 9950: 9936: 9928: 9886: 9878: 9855: 9847: 9808: 9800: 9757: 9749: 9715: 9707: 9679: 9671: 9608: 9600: 9577: 9569: 9472: 9464: 9433: 9427: 9413: 9361: 9344: 9312: 9304: 9284: 9276: 9242: 9234: 9212: 9204: 8795: 8787: 8394: 8386: 8021: 8013: 7978: 7970: 7821:are such that the action integral 7740: 7725: 7693: 7685: 7573: 7560: 7196: 7137: 7130: 7125: 7096: 7089: 7084: 7069: 7045: 6883: 6846: 6772: 6744: 6707: 6676: 6340: 6268: 6180: 6157: 6150: 6145: 6125: 6043: 6033: 6014: 6004: 5967: 5858: 5848: 5836: 5826: 5799: 5775: 5753: 5686: 5660: 5635:can be written infinitesimally as 5547: 5530: 5517: 5273: 5265: 5013: 5005: 4851: 4843: 4725: 4717: 4653: 4645: 4339: 4296: 4140: 4132: 4022: 4014: 3945: 3937: 3907: 3899: 3451: 3401: 3378: 3322: 3296: 2989: 2978: 2878: 2870: 2820: 2795: 2787: 2755: 2731: 2642: 2631: 2556:undergo a symmetry transformation 2485: 2474: 2409: 2398: 2331: 2323: 2270: 2262: 2231: 2134: 2126: 2093: 2085: 2027: 2019: 1995: 1977: 1969: 1954: 1942: 1910: 1888: 1877: 1861: 1513: 1107:within this system, respectively. 1074:Basic illustrations and background 54:Part of a series of articles about 32:Noether's theorem (disambiguation) 25: 23852: 23568:"Noether's Theorem in a Nutshell" 23480: 23063:. Boston, MA: Birkhäuser Boston. 23023:. Vol. 1. Wiley. p. 5. 22857: 19038:Example 1: Conservation of energy 5494:{\displaystyle T(\mathbf {x} ,t)} 5111:; such a rotation transforms the 1194:. The Noether current is defined 1134:Informal statement of the theorem 23723:Emmy Noether's Wonderful Theorem 23435:. Vol. 107 (2nd ed.). 22775: 22761: 15579:{\displaystyle \mathbb {R} ^{3}} 15546:{\displaystyle \mathbb {R} ^{m}} 15422:is the one-dimensional manifold 15331:Manifold/fiber bundle derivation 12021: 11976: 11959: 11903: 11831: 11786: 11775: 11747: 11645: 11575: 11516: 11505: 11477: 11427: 11416: 11388: 11322: 11295: 11240: 11213: 11156: 11145: 11104: 11056: 11028: 11017: 10976: 10917: 10838: 10753: 10742: 10701: 10653: 10568: 10557: 10529: 10479: 10468: 10440: 10374: 10347: 10292: 10265: 10163: 10152: 10124: 10090: 10071: 10050: 10007: 9973: 9962: 9940: 9901: 9890: 9862: 9828: 9812: 9764: 9730: 9719: 9686: 9623: 9612: 9584: 9487: 9476: 9448: 9437: 9387: 9365: 9319: 9257: 9246: 9216: 9115: 9088: 9033: 9006: 8930:. Calculating the derivative at 8850: 8810: 8799: 8745: 8621: 8591: 8449: 8409: 8398: 8358: 8305: 8278: 8203: 8171: 8139: 8117: 8025: 7985: 7892: 7872: 5927:{\displaystyle r=1,2,\ldots ,N,} 5478: 5381: 5373: 5360: 5352: 5339: 5326: 5318: 5305: 5297: 5280: 5232: 5224: 5216: 5157: 5149: 5134: 5126: 4878: 4858: 4750: 4732: 4680: 4660: 4612:, and the parameters make up an 4514: 4481: 4359: 4348: 4334: 4321: 3949: 3914: 3814: 3794: 3727: 3716: 3679:is defined as the time integral 1213:stipulates for the invariants: 23489:"Invariante Variationsprobleme" 23215: 22843:"Invariante Variationsprobleme" 22660: 21469:{\displaystyle {\vec {x}}_{CM}} 19369:{\displaystyle Q={\dot {x}}(t)} 7651: = 0) current equals 7507:becomes infinitesimally small, 5197:as the generalized coordinates 4591:of the generalized coordinates. 4245:Simple form using perturbations 3649:of astronomical bodies, is the 1854:That changes the Lagrangian by 1440:The total change in the action 1312:(gray arrows on the diagram). 1205:In the context of gravitation, 1202:(divergenceless) vector field. 23826:Partial differential equations 23085: 23037: 23010: 22963: 22930: 22895: 22834: 22603:Noether's theorem states that 22430: 22424: 22403: 22387: 21963: 21953: 21901: 21895: 21886: 21880: 21851: 21848: 21842: 21836: 21693: 21687: 21668: 21662: 21634: 21628: 21551: 21530: 21521: 21498: 21451: 21417: 21372: 21350: 21317: 21277: 21227: 21202: 21185: 21144: 21098: 21050: 21014: 20840: 20818: 20707: 20697: 20628: 20622: 20566: 20482: 20460: 20389: 20305: 20299: 20288: 20274: 20268: 20262: 20224: 20010: 20004: 19936: 19930: 19816: 19803: 19700: 19694: 19575: 19569: 19485: 19479: 19403: 19397: 19363: 19357: 19339: 19336: 19330: 19324: 19256: 19253: 19247: 19241: 19147: 19141: 19123: 19117: 19093: 19087: 18993: 18987: 18966: 18940: 18905: 18899: 18868: 18842: 18819: 18813: 18792: 18776: 18662: 18656: 18635: 18619: 18539: 18533: 18459: 18449: 18404: 18401: 18395: 18386: 18358: 18342: 18322: 18316: 18244: 18234: 18231: 18225: 18209: 18199: 18196: 18190: 18134: 18124: 18056: ≈ 0 for every 17964: 17961: 17958: 17952: 17939: 17923: 17920: 17914: 17901: 17888: 17870: 17854: 17753: 17750: 17740: 17727: 17711: 17708: 17698: 17685: 17669: 17659: 17656: 17630: 17580: 17570: 17512: 17502: 17467:Generalization to Lie algebras 17256: 17250: 17241: 17225: 17111: 17105: 17096: 17080: 16945: 16939: 16930: 16914: 16861: 16855: 16841: 16825: 16620: 16611: 16605: 16586: 16580: 16574: 16561: 16555: 16518: 16512: 16486: 16483: 16477: 16467: 16420: 16390: 16384: 16378: 16278:{\displaystyle {\mathcal {C}}} 16252:principle of stationary action 16201: 16195: 16184: 16178: 16128:{\displaystyle {\mathcal {S}}} 16080:{\displaystyle {\mathcal {C}}} 15966: 15957: 15951: 15932: 15926: 15920: 15895: 15889: 15866:{\displaystyle {\mathcal {C}}} 15785: 15757: 15725:{\displaystyle {\mathcal {S}}} 15719: 15713: 15620: 15524:, then the target manifold is 15368:{\displaystyle {\mathcal {C}}} 14992: 14667: 14449: 14233: 14100: 13997: 13933: 13537: 13461: 13424: 13360: 13296: 12449: 12220: 12145: 11691: 11678: 11621: 11608: 11539: 11526: 11450: 11437: 11361: 11345: 11332: 11312: 11299: 11291: 11279: 11263: 11250: 11230: 11217: 11209: 10591: 10578: 10502: 10489: 10413: 10397: 10384: 10364: 10351: 10343: 10331: 10315: 10302: 10282: 10269: 10261: 10248: 10242: 10055: 10043: 9370: 9358: 9154: 9138: 9125: 9105: 9092: 9084: 9072: 9056: 9043: 9023: 9010: 9002: 8989: 8983: 8894: 8880: 8860: 8843: 8834: 8814: 8806: 8778: 8769: 8749: 8741: 8735: 8660: 8646: 8635: 8610: 8599: 8586: 8526: 8520: 8479: 8459: 8442: 8433: 8413: 8405: 8377: 8368: 8362: 8354: 8330: 8319: 8297: 8294: 8288: 8236: 8227: 8207: 8199: 8190: 8181: 8175: 8167: 8158: 8147: 8134: 8127: 8121: 8086: 8038: 8032: 8004: 7998: 7917: 7908: 7902: 7882: 7876: 7868: 7807: 7800:, and is one of the prototype 7455: 7417: 7336:Canonical stress–energy tensor 6767: 6504: 6501: 6488: 6403: 6206:is understood to contract the 6009: 5947: 5831: 5807:{\displaystyle {\mathcal {L}}} 5783:{\displaystyle {\mathcal {S}}} 5648: 5488: 5474: 5130: 4329: 4288: 3851:states that the physical path 3743: 3712: 3338:is interpreted as a flux of a 3219: 3200: 3160: 3134: 3117:{\displaystyle \psi (q_{r},t)} 3111: 3092: 3069: 3063: 2702: 2573: 2370: 2357: 2309: 2296: 1809: 1700:{\displaystyle \delta q/\tau } 1493: 1467: 1331: 1325: 1287: 139: 133: 124: 118: 102: 96: 13: 1: 23821:Eponymous theorems of physics 23433:Graduate Texts in Mathematics 23403:Moser, Seth (21 April 2020). 23288:: CS1 maint: date and year ( 22828: 22687:law of conservation of energy 19032: 16688:(and only on-shell), we have 16450:for all compact submanifolds 8498:The action integral flows to 5075:at the space-time point: the 1060:Rayleigh dissipation function 1042:Noether's theorem is used in 1019:) published by mathematician 439:Integral of inverse functions 22807:Gauge symmetry (mathematics) 19299:. Consider the generator of 18714:{\displaystyle \varepsilon } 16259:infinitesimal transformation 15685:{\displaystyle \mathbb {C} } 15663:{\displaystyle \mathbb {R} } 15437:{\displaystyle \mathbb {R} } 14826:{\displaystyle \varphi ^{A}} 14795:{\displaystyle \varphi ^{A}} 12098:{\displaystyle \varphi ^{A}} 11714:From which one can see that 6595:with the value at the point 4911:II. Translational invariance 4604:is a constant with units of 2717:{\displaystyle t\mapsto t+T} 1037:symmetries of physical space 7: 23621:American Journal of Physics 23374:Kosmann-Schwarzbach, Yvette 23201:. Basic Books. p. 18. 22754: 18004:Generalization of the proof 17459:quantities as well are the 17315: 17033:Since this is true for any 4780: 3683:of a function known as the 3673:Cartesian coordinate system 857:Calculus on Euclidean space 280:Logarithmic differentiation 10: 23857: 23678:Revista Mexicana de Física 23608:10.1103/PhysRevD.76.025025 21426:{\displaystyle {\vec {P}}} 20575:{\displaystyle {\vec {Q}}} 18012:local symmetry derivation 12074:Field-theoretic derivation 6993:, Noether's theorem gives 5409:axis is conserved. And if 5083:III. Rotational invariance 5066:(as it is done above), or 4237: 3660:, which is related to the 3495:states that some quantity 3476: 3460:{\displaystyle \partial U} 3387:{\displaystyle \partial U} 3305:{\displaystyle \partial U} 3050:to the conserved quantity. 2521:, the canonical momentum. 1669:{\displaystyle {\dot {q}}} 1640:{\displaystyle {\dot {q}}} 1271:and a continuous symmetry 1178:is usually expressed as a 1007:of a physical system with 29: 23789:10.1142/S0219887809003862 23528:10.1080/00411457108231446 23409:Physics Capstone Projects 23349:Johnson, Tristan (2016). 23324:10.1007/978-3-319-59695-2 23310:. Springer. p. 218. 23104:10.1007/978-3-030-63810-8 23069:10.1007/978-1-4684-0535-4 22996:10.1142/S0217732386000385 22649:If one tries to find the 17461:Ward–Takahashi identities 15452:of generalized positions. 6478:), affects the fields as 5503:principle of least action 4619:Using these definitions, 4212:canonical transformations 4181:Thus, the absence of the 3662:principle of least action 3651:Laplace–Runge–Lenz vector 3225:{\displaystyle (q_{r},t)} 1371:governing this system is 1124:Ward–Takahashi identities 1033:principle of least action 591:Summand limit (term test) 23260:10.1103/PhysRevD.52.4430 22976:Modern Physics Letters A 21933:{\displaystyle \varphi } 21597:conformal transformation 19384:does not; consequently: 19054:, coordinatized by time 18104:{\displaystyle \varphi } 18068:) ≈ 0 for all 17320:Noether's theorem is an 17175:{\displaystyle J^{\mu }} 16257:Now, suppose we have an 16239:{\displaystyle \varphi } 16148:{\displaystyle \varphi } 16100:{\displaystyle \varphi } 16087:consisting of functions 16048:{\displaystyle \varphi } 16016:{\displaystyle \varphi } 15842:{\displaystyle \varphi } 15818:{\displaystyle \varphi } 15648:. (It takes values into 12805:into the following form 7941:Euler–Lagrange equations 7813:One independent variable 7533:relativistically correct 7376:, because it ascribes a 6658:{\displaystyle x^{\mu }} 6588:{\displaystyle x^{\mu }} 5739:{\displaystyle \varphi } 5719:{\displaystyle x^{\mu }} 5628:{\displaystyle \varphi } 5457:{\displaystyle \varphi } 5417:is conserved; in short, 5087:The conservation of the 4403:symmetry transformations 4386:where the perturbations 4217:Hamilton–Jacobi equation 3865:Euler–Lagrange equations 3410:{\displaystyle \Delta S} 3331:{\displaystyle \Delta S} 3315:and its contribution to 2740:{\displaystyle \Delta S} 2240:{\displaystyle \Delta S} 1723:{\displaystyle \delta q} 1542:{\displaystyle \varphi } 1522:{\displaystyle \Delta S} 1392:{\displaystyle \varphi } 1315:Consider any trajectory 1017:Noether's second theorem 275:Implicit differentiation 265:Differentiation notation 192:Inverse function theorem 23092:Rowe, David E. (2021). 23017:Thompson, W.J. (1994). 22714:Ward–Takahashi identity 21433:is the total momentum, 20066:{\displaystyle dj/dt=0} 16059:approaches ∞. Then the 15590:Now suppose there is a 7307:{\displaystyle \sigma } 6467:{\displaystyle \delta } 5205:variables are given by 4262:generalized coordinates 4234:Mathematical expression 3666:generalized coordinates 3623:Newton's laws of motion 733:Helmholtz decomposition 23806:Calculus of variations 23060:Emmy Noether 1882–1935 23057:Dick, Auguste (1981). 22749:stationary black holes 22640: 22594: 22333: 22277: 22122: 21934: 21911: 21810: 21581: 21470: 21437:is the total mass and 21427: 21395: 21069: 20992: 20922: 20664: 20576: 20544: 20521: 20357: 20172: 20106: 20067: 20024: 19763: 19710: 19619: 19370: 19282: 19210: 19023: 18715: 18692: 18164: 18105: 17992: 17788: 17615: 17547: 17449: 17381: 17279: 17176: 17142: 17024: 16630: 16525: 16441: 16279: 16240: 16217: 16149: 16129: 16109:functional derivatives 16101: 16081: 16049: 16017: 15990: 15867: 15843: 15819: 15792: 15726: 15686: 15664: 15635: 15580: 15547: 15518: 15438: 15369: 15347:and a target manifold 15339:-dimensional oriented 15321: 15131: 15062: 14967: 14872: 14827: 14796: 14765: 14721: 14561: 14358: 14132: 13688: 13566: 13396: 13106: 12792: 12706: 12491: 12357: 12324: 12191: 12099: 12061: 11920: 11868: 11705: 11173: 10781: 10188: 9523: 8917: 8489: 8247: 8048: 7931: 7787: 7638: 7494: 7328: 7308: 7288: 7265: 7205: 6987: 6967: 6947: 6927: 6892: 6829: 6726: 6659: 6632: 6589: 6562: 6468: 6448: 6376: 6300: 6244: 6224: 6193: 6087: 6064: 5983: 5928: 5874: 5808: 5784: 5760: 5740: 5720: 5693: 5670: 5629: 5595: 5495: 5458: 5392: 5243: 5168: 5045: 4901: 4764: 4538: 4377: 4208:William Rowan Hamilton 4169: 4097: 3966: 3839: 3763: 3576: 3461: 3438: 3411: 3388: 3362: 3332: 3306: 3283: 3256: 3226: 3187: 3173:with a bounded region 3167: 3118: 3076: 3044: 2934: 2741: 2718: 2683: 2603: 2550: 2529:When more coordinates 2515: 2456: 2436: 2380: 2241: 2218: 2191: 2162: 1935:, which integrates to 1929: 1845: 1784: 1744: 1724: 1701: 1670: 1641: 1612: 1592: 1563: 1543: 1523: 1500: 1454: 1431: 1393: 1365: 1338: 1306: 1265: 1248: 1236: 1160: 1144: 1048:calculus of variations 867:Limit of distributions 687:Directional derivative 348:Faà di Bruno's formula 146: 48: 23487:Emmy Noether (1918). 22641: 22595: 22334: 22278: 22131:This has the form of 22123: 21935: 21912: 21811: 21603:in (3 + 1)- 21582: 21471: 21428: 21396: 21070: 20993: 20931:This has the form of 20923: 20665: 20577: 20545: 20522: 20337: 20173: 20086: 20068: 20025: 19743: 19711: 19620: 19371: 19283: 19190: 19024: 18716: 18693: 18165: 18106: 18060:, or more compactly, 17993: 17789: 17616: 17548: 17450: 17382: 17280: 17177: 17143: 17025: 16631: 16526: 16442: 16280: 16241: 16218: 16150: 16130: 16102: 16082: 16050: 16018: 15999:Suppose we are given 15991: 15868: 15844: 15820: 15793: 15727: 15687: 15665: 15636: 15581: 15548: 15519: 15439: 15370: 15322: 15132: 15063: 14968: 14873: 14828: 14797: 14766: 14722: 14562: 14359: 14133: 13689: 13567: 13397: 13107: 12793: 12707: 12492: 12358: 12325: 12192: 12100: 12062: 11921: 11869: 11706: 11174: 10782: 10189: 9524: 8918: 8490: 8248: 8049: 7932: 7788: 7639: 7529:Klein–Gordon equation 7495: 7404:| unchanged, such as 7359:probability amplitude 7349:linear in the fields 7329: 7309: 7289: 7266: 7206: 6988: 6968: 6948: 6928: 6913:, where we have used 6893: 6830: 6727: 6660: 6633: 6590: 6563: 6469: 6449: 6377: 6301: 6245: 6225: 6194: 6088: 6065: 5992:with the consequence 5984: 5929: 5875: 5809: 5785: 5761: 5759:{\displaystyle \Psi } 5741: 5721: 5694: 5692:{\displaystyle \Psi } 5671: 5630: 5596: 5496: 5459: 5393: 5244: 5169: 5113:Cartesian coordinates 5046: 4902: 4765: 4539: 4378: 4170: 4098: 3967: 3840: 3764: 3577: 3462: 3439: 3437:{\displaystyle j_{r}} 3412: 3389: 3363: 3361:{\displaystyle j_{r}} 3333: 3307: 3284: 3282:{\displaystyle t_{1}} 3257: 3255:{\displaystyle t_{0}} 3227: 3188: 3168: 3119: 3077: 3045: 2935: 2742: 2719: 2684: 2604: 2551: 2549:{\displaystyle q_{r}} 2516: 2457: 2437: 2386:meaning the quantity 2381: 2242: 2219: 2217:{\displaystyle t_{1}} 2192: 2190:{\displaystyle t_{0}} 2163: 1930: 1846: 1785: 1745: 1743:{\displaystyle \tau } 1725: 1702: 1671: 1642: 1613: 1593: 1591:{\textstyle S=\int L} 1564: 1544: 1524: 1501: 1455: 1432: 1430:{\displaystyle \tau } 1394: 1366: 1339: 1307: 1266: 1246: 1215: 1148: 1140: 1068:continuous symmetries 1056:Hamiltonian mechanics 951:Mathematical analysis 862:Generalized functions 547:arithmetico-geometric 393:Leibniz integral rule 147: 42: 23836:Quantum field theory 22905:Analytical Mechanics 22841:Noether, E. (1918). 22710:quantum field theory 22607: 22349: 22293: 22138: 21947: 21924: 21830: 21614: 21483: 21441: 21408: 21085: 21005: 20935: 20680: 20589: 20557: 20534: 20200: 20083: 20037: 19726: 19635: 19391: 19318: 19073: 18731: 18705: 18180: 18118: 18095: 17819: 17627: 17557: 17489: 17394: 17332: 17300:this current over a 17292:associated with the 17288:which is called the 17189: 17159: 17044: 16695: 16663:is a generator of a 16545: 16461: 16303: 16265: 16230: 16162: 16139: 16115: 16091: 16067: 16039: 16007: 15879: 15853: 15833: 15809: 15747: 15703: 15674: 15652: 15600: 15561: 15528: 15482: 15426: 15406:in physics include: 15355: 15147: 15086: 14983: 14885: 14837: 14810: 14779: 14734: 14581: 14374: 14148: 13704: 13582: 13415: 13122: 12812: 12722: 12515: 12378: 12337: 12207: 12132: 12082: 11933: 11885: 11721: 11189: 10797: 10204: 9539: 8945: 8505: 8271: 8072: 7949: 7827: 7658: 7550: 7537:Schrödinger equation 7411: 7341:The conservation of 7318: 7298: 7287:{\displaystyle \mu } 7278: 7221: 7000: 6977: 6966:{\displaystyle \mu } 6957: 6937: 6926:{\displaystyle \mu } 6917: 6903:stress–energy tensor 6842: 6739: 6672: 6642: 6599: 6572: 6482: 6458: 6390: 6320: 6264: 6234: 6223:{\displaystyle \nu } 6214: 6103: 6077: 5999: 5941: 5891: 5821: 5794: 5770: 5750: 5746:. The condition for 5730: 5703: 5683: 5642: 5619: 5512: 5468: 5448: 5425:Field theory version 5259: 5212: 5201:, the corresponding 5122: 5077:stress–energy tensor 4986: 4959: = 0, and 4831: 4634: 4423: 4274: 4113: 3989: 3874: 3849:Hamilton's principle 3787: 3697: 3658:Lagrangian mechanics 3643:stress–energy tensor 3631:stress–energy tensor 3522: 3448: 3421: 3398: 3375: 3345: 3319: 3293: 3266: 3239: 3197: 3177: 3131: 3086: 3075:{\displaystyle q(t)} 3057: 2946: 2752: 2728: 2696: 2613: 2560: 2533: 2466: 2446: 2390: 2251: 2228: 2201: 2174: 1939: 1858: 1794: 1754: 1734: 1711: 1680: 1651: 1622: 1602: 1573: 1553: 1533: 1510: 1464: 1444: 1421: 1383: 1355: 1337:{\displaystyle q(t)} 1319: 1275: 1255: 1150:To every continuous 1011:has a corresponding 956:Nonstandard analysis 429:Lebesgue integration 299:Rules and identities 70: 23816:Concepts in physics 23781:2009arXiv0906.1732S 23700:2006RMxF...52...29M 23633:2004AmJPh..72..428H 23600:2007PhRvD..76b5025C 23520:1971TTSP....1..186N 23316:2018vffg.book.....B 23252:1995PhRvD..52.4430I 23171:Classical Mechanics 22988:1986MPLA....1..293D 22812:Invariant (physics) 21605:Minkowski spacetime 20987: 20910: 20760: 20656: 20621: 19997: 19923: 19871: 19687: 19472: 19234: 17984: 17836: 17783: 17610: 17542: 17153:continuity equation 16454:or in other words, 16155:are zero, that is: 16035:, or some limit on 16001:boundary conditions 15412:classical mechanics 15377:configuration space 15341:Riemannian manifold 15335:Suppose we have an 10637: 9200: 8731: 8582: 7864: 7372:is a complex field 7193: 7067: 7042: 6880: 6859: 6819: 6705: 6552: 6443: 6253:In such cases, the 6138: 6120: 6056: 4814: = 1 and 4775:constants of motion 4240:Perturbation theory 4106:where the momentum 3772:where the dot over 3647:celestial mechanics 3587:constants of motion 1180:continuity equation 1052:constants of motion 1044:theoretical physics 1009:conservative forces 1001:continuous symmetry 627:Cauchy condensation 434:Contour integration 160:Fundamental theorem 87: 23166:Goldstein, Herbert 23043:Nina Byers (1998) 22822:Symmetry (physics) 22769:Mathematics portal 22734:electric potential 22636: 22590: 22588: 22329: 22273: 22118: 21930: 21907: 21806: 21804: 21577: 21466: 21423: 21391: 21389: 21252: 21120: 21065: 21032: 20988: 20973: 20962: 20918: 20916: 20887: 20876: 20779: 20737: 20726: 20660: 20642: 20607: 20572: 20540: 20517: 20515: 20434: 20168: 20063: 20020: 20018: 19974: 19973: 19900: 19899: 19848: 19847: 19706: 19664: 19663: 19615: 19559: 19508: 19449: 19448: 19366: 19314:. In other words, 19278: 19276: 19211: 19019: 18711: 18688: 18686: 18160: 18101: 17988: 17970: 17822: 17784: 17769: 17611: 17596: 17543: 17528: 17445: 17377: 17306:conserved quantity 17275: 17172: 17138: 17020: 17018: 16626: 16521: 16437: 16275: 16236: 16213: 16145: 16125: 16097: 16077: 16045: 16013: 15986: 15863: 15839: 15815: 15803:Lagrangian density 15788: 15722: 15682: 15660: 15631: 15576: 15543: 15514: 15434: 15365: 15317: 15127: 15058: 14963: 14868: 14823: 14792: 14761: 14717: 14715: 14557: 14354: 14128: 14126: 13684: 13562: 13392: 13102: 12803:divergence theorem 12788: 12702: 12487: 12353: 12320: 12187: 12117:ranges over time ( 12095: 12057: 11916: 11864: 11701: 11699: 11169: 10777: 10775: 10609: 10184: 10182: 9519: 9517: 9172: 8913: 8911: 8685: 8536: 8485: 8243: 8241: 8044: 7927: 7836: 7783: 7634: 7490: 7324: 7304: 7284: 7261: 7201: 7179: 7053: 7028: 6983: 6963: 6943: 6923: 6888: 6866: 6845: 6825: 6805: 6722: 6691: 6655: 6628: 6585: 6558: 6538: 6464: 6444: 6429: 6372: 6296: 6240: 6220: 6189: 6124: 6106: 6083: 6060: 6042: 5979: 5924: 5885:divergence theorem 5870: 5804: 5780: 5756: 5736: 5716: 5689: 5666: 5625: 5591: 5491: 5454: 5388: 5239: 5164: 5060:general relativity 5041: 4897: 4786:I. Time invariance 4760: 4595:For translations, 4534: 4532: 4501: 4452: 4373: 4371: 4165: 4093: 3962: 3835: 3759: 3627:general relativity 3572: 3479:Constant of motion 3473:Historical context 3457: 3434: 3407: 3384: 3358: 3328: 3302: 3289:with the boundary 3279: 3252: 3222: 3183: 3163: 3114: 3072: 3040: 2972: 2930: 2737: 2714: 2679: 2625: 2599: 2546: 2511: 2452: 2432: 2376: 2237: 2214: 2187: 2158: 1925: 1841: 1780: 1740: 1720: 1697: 1666: 1637: 1608: 1588: 1559: 1539: 1519: 1496: 1450: 1427: 1399:on a time segment 1389: 1361: 1334: 1302: 1261: 1249: 1062:). In particular, 1054:in Lagrangian and 1025:integral over time 999:states that every 799:Partial derivative 728:generalized Stokes 622:Alternating series 503:Reduction formulae 478:tangent half-angle 465:Cylindrical shells 388:Integral transform 383:Lists of integrals 187:Mean value theorem 142: 73: 49: 23811:Conservation laws 23732:978-0-8018-9694-1 23663:Noether's Theorem 23641:10.1119/1.1591764 23588:Physical Review D 23472:978-94-6239-171-0 23455:Sardanashvily, G. 23391:978-0-387-87867-6 23333:978-3-319-59694-5 23229:Physical Review D 23113:978-3-030-63809-2 23078:978-1-4684-0537-8 22948:978-0-534-40896-1 22880:978-1-139-64890-5 22534: 22407: 22164: 21740: 21601:quartic potential 21559: 21554: 21533: 21513: 21501: 21454: 21420: 21375: 21353: 21320: 21285: 21280: 21243: 21230: 21205: 21188: 21162: 21152: 21147: 21111: 21101: 21053: 21023: 21017: 20953: 20951: 20897: 20867: 20843: 20821: 20764: 20747: 20717: 20569: 20543:{\displaystyle N} 20485: 20463: 20419: 20397: 20392: 20373: 20296: 20291: 20265: 20227: 20159: 20141: 20131: 19984: 19964: 19962: 19910: 19890: 19888: 19858: 19838: 19798: 19788: 19674: 19654: 19652: 19606: 19594: 19550: 19538: 19519: 19499: 19459: 19439: 19437: 19422: 19376:. The coordinate 19354: 19301:time translations 19221: 19188: 19138: 18970: 18872: 18796: 18639: 18516: 18362: 18296: 17874: 17431: 17420: 17245: 17100: 16934: 16845: 16792: 16680:, because of the 16285:, generated by a 16205: 15402:Examples of this 15297: 15205: 15109: 14995: 14947: 14670: 14452: 14441: 14397: 14236: 14225: 14181: 14103: 14092: 14048: 14000: 13989: 13936: 13921: 13878: 13682: 13648: 13605: 13540: 13524: 13499: 13464: 13427: 13363: 13352: 13299: 13288: 12988: 12782: 12452: 12052: 12032: 12028: 11983: 11970: 11966: 11908: 11862: 11842: 11838: 11793: 11780: 11758: 11754: 11676: 11656: 11652: 11606: 11586: 11582: 11523: 11510: 11488: 11484: 11434: 11421: 11399: 11395: 11329: 11247: 11163: 11150: 11115: 11111: 11081: 11061: 11035: 11022: 10987: 10983: 10953: 10928: 10924: 10897: 10869: 10849: 10845: 10813: 10760: 10747: 10712: 10708: 10678: 10658: 10575: 10562: 10540: 10536: 10486: 10473: 10451: 10447: 10381: 10299: 10240: 10170: 10157: 10135: 10131: 10097: 10078: 10065: 10018: 10014: 9980: 9967: 9945: 9908: 9895: 9873: 9869: 9835: 9817: 9795: 9775: 9771: 9737: 9724: 9697: 9693: 9666: 9630: 9617: 9595: 9591: 9559: 9494: 9481: 9455: 9442: 9394: 9380: 9330: 9326: 9291: 9264: 9251: 9221: 9122: 9040: 8981: 8857: 8804: 8628: 8456: 8403: 8349: 8312: 8285: 8030: 7996: 7992: 7965: 7899: 7779: 7754: 7707: 7486: 7444: 7438: 7394:complex conjugate 7355:quantum mechanics 7327:{\displaystyle T} 7154: 7113: 6986:{\displaystyle T} 6946:{\displaystyle r} 6243:{\displaystyle r} 6210:indices, not the 6174: 6095:current densities 6086:{\displaystyle N} 5291: 5287: 5036: 5032: 4885: 4869: 4865: 4743: 4739: 4687: 4671: 4667: 4526: 4492: 4443: 4365: 4163: 4153: 4089: 4079: 4045: 4035: 4005: 3958: 3954: 3925: 3921: 3890: 3831: 3827: 3801: 3755: 3734: 3611:Gottfried Leibniz 3589:(although motion 3568: 3557: 3543: 3487:conserved current 3340:conserved current 3186:{\displaystyle U} 3026: 3002: 2963: 2913: 2901: 2891: 2833: 2818: 2808: 2655: 2616: 2497: 2455:{\displaystyle q} 2421: 2347: 2343: 2286: 2282: 2150: 2146: 2109: 2105: 2078: 2066: 2043: 2039: 2007: 1993: 1989: 1922: 1900: 1821: 1806: 1707:, and the change 1663: 1634: 1611:{\displaystyle q} 1562:{\displaystyle L} 1453:{\displaystyle S} 1364:{\displaystyle S} 1264:{\displaystyle q} 1230:parameters, then 1176:physical quantity 1156:conserved current 1097:conservation laws 997:Noether's theorem 994: 993: 874: 873: 836: 835: 804:Multiple integral 740: 739: 644: 643: 611:Direct comparison 582:Convergence tests 520: 519: 493:Partial fractions 360: 359: 270:Second derivative 16:(Redirected from 23848: 23831:Physics theorems 23792: 23774: 23765:(6): 1047–1056. 23751: 23749: 23736: 23717: 23715: 23714: 23708: 23702:. Archived from 23693: 23675: 23659: 23657: 23644: 23611: 23582: 23580: 23578: 23559: 23557: 23539: 23513: 23492: 23476: 23450: 23420: 23418: 23416: 23395: 23369: 23367: 23365: 23345: 23294: 23293: 23287: 23279: 23245: 23236:(8): 4430–4439. 23219: 23213: 23212: 23192: 23186: 23185: 23162: 23151: 23150: 23129: 23118: 23117: 23089: 23083: 23082: 23054: 23048: 23041: 23035: 23034: 23014: 23008: 23007: 22967: 22961: 22960: 22934: 22928: 22927: 22899: 22893: 22892: 22864: 22855: 22854: 22838: 22797:Charge (physics) 22792:Conservation law 22785: 22780: 22779: 22771: 22766: 22765: 22738:vector potential 22698:angular momentum 22645: 22643: 22642: 22637: 22629: 22628: 22619: 22618: 22599: 22597: 22596: 22591: 22589: 22582: 22578: 22577: 22576: 22558: 22557: 22545: 22544: 22535: 22527: 22520: 22519: 22507: 22503: 22493: 22492: 22483: 22482: 22465: 22464: 22449: 22445: 22444: 22420: 22416: 22415: 22414: 22408: 22406: 22399: 22398: 22379: 22365: 22364: 22338: 22336: 22335: 22330: 22325: 22324: 22318: 22317: 22305: 22304: 22282: 22280: 22279: 22274: 22272: 22268: 22267: 22266: 22260: 22259: 22245: 22244: 22232: 22228: 22227: 22226: 22217: 22216: 22198: 22197: 22185: 22184: 22175: 22174: 22165: 22157: 22150: 22149: 22127: 22125: 22124: 22119: 22114: 22110: 22100: 22099: 22090: 22089: 22075: 22074: 22056: 22052: 22048: 22047: 22032: 22031: 22022: 22021: 22012: 22011: 21996: 21995: 21978: 21977: 21962: 21961: 21939: 21937: 21936: 21931: 21916: 21914: 21913: 21908: 21876: 21875: 21866: 21865: 21815: 21813: 21812: 21807: 21805: 21798: 21797: 21788: 21784: 21783: 21782: 21764: 21763: 21751: 21750: 21741: 21733: 21717: 21710: 21709: 21700: 21696: 21683: 21682: 21653: 21652: 21627: 21626: 21586: 21584: 21583: 21578: 21570: 21569: 21561: 21560: 21555: 21547: 21545: 21535: 21534: 21526: 21514: 21512: 21504: 21503: 21502: 21494: 21487: 21475: 21473: 21472: 21467: 21465: 21464: 21456: 21455: 21447: 21432: 21430: 21429: 21424: 21422: 21421: 21413: 21400: 21398: 21397: 21392: 21390: 21386: 21385: 21377: 21376: 21368: 21355: 21354: 21346: 21337: 21333: 21329: 21328: 21327: 21322: 21321: 21313: 21309: 21308: 21293: 21292: 21287: 21286: 21281: 21273: 21271: 21267: 21266: 21251: 21236: 21232: 21231: 21223: 21217: 21213: 21212: 21207: 21206: 21198: 21190: 21189: 21181: 21175: 21171: 21170: 21169: 21163: 21161: 21160: 21159: 21154: 21153: 21148: 21140: 21138: 21127: 21119: 21103: 21102: 21094: 21074: 21072: 21071: 21066: 21061: 21060: 21055: 21054: 21046: 21042: 21041: 21031: 21019: 21018: 21010: 20997: 20995: 20994: 20989: 20986: 20981: 20972: 20971: 20961: 20952: 20950: 20939: 20927: 20925: 20924: 20919: 20917: 20909: 20904: 20899: 20898: 20890: 20886: 20885: 20875: 20860: 20856: 20852: 20851: 20850: 20845: 20844: 20836: 20829: 20828: 20823: 20822: 20814: 20805: 20804: 20792: 20791: 20778: 20759: 20754: 20749: 20748: 20740: 20736: 20735: 20725: 20706: 20705: 20696: 20695: 20669: 20667: 20666: 20661: 20655: 20650: 20635: 20631: 20620: 20615: 20601: 20600: 20581: 20579: 20578: 20573: 20571: 20570: 20562: 20549: 20547: 20546: 20541: 20526: 20524: 20523: 20518: 20516: 20503: 20499: 20498: 20494: 20493: 20492: 20487: 20486: 20478: 20471: 20470: 20465: 20464: 20456: 20447: 20446: 20433: 20415: 20414: 20409: 20405: 20404: 20399: 20398: 20393: 20385: 20383: 20374: 20369: 20368: 20359: 20356: 20351: 20322: 20312: 20308: 20298: 20297: 20292: 20284: 20282: 20267: 20266: 20258: 20250: 20249: 20233: 20229: 20228: 20220: 20213: 20212: 20185:) is conserved. 20177: 20175: 20174: 20169: 20161: 20160: 20155: 20154: 20145: 20142: 20140: 20139: 20138: 20133: 20132: 20124: 20116: 20108: 20105: 20100: 20072: 20070: 20069: 20064: 20050: 20029: 20027: 20026: 20021: 20019: 19996: 19991: 19986: 19985: 19977: 19972: 19963: 19955: 19947: 19943: 19939: 19922: 19917: 19912: 19911: 19903: 19898: 19889: 19881: 19870: 19865: 19860: 19859: 19851: 19846: 19828: 19815: 19814: 19799: 19797: 19796: 19795: 19790: 19789: 19781: 19773: 19765: 19762: 19757: 19715: 19713: 19712: 19707: 19686: 19681: 19676: 19675: 19667: 19662: 19653: 19645: 19624: 19622: 19621: 19616: 19614: 19613: 19608: 19607: 19599: 19595: 19593: 19592: 19591: 19578: 19561: 19558: 19546: 19545: 19540: 19539: 19531: 19527: 19526: 19521: 19520: 19512: 19507: 19492: 19488: 19471: 19466: 19461: 19460: 19452: 19447: 19438: 19430: 19423: 19421: 19410: 19375: 19373: 19372: 19367: 19356: 19355: 19347: 19297:potential energy 19287: 19285: 19284: 19279: 19277: 19263: 19259: 19233: 19228: 19223: 19222: 19214: 19209: 19204: 19189: 19181: 19165: 19154: 19150: 19140: 19139: 19131: 19086: 19085: 19028: 19026: 19025: 19020: 19012: 19008: 19000: 18996: 18983: 18979: 18978: 18977: 18971: 18969: 18962: 18961: 18952: 18951: 18932: 18920: 18919: 18895: 18894: 18885: 18881: 18880: 18879: 18873: 18871: 18864: 18863: 18854: 18853: 18834: 18809: 18805: 18804: 18803: 18797: 18795: 18788: 18787: 18768: 18758: 18757: 18743: 18742: 18720: 18718: 18717: 18712: 18697: 18695: 18694: 18689: 18687: 18680: 18679: 18669: 18665: 18652: 18648: 18647: 18646: 18640: 18638: 18631: 18630: 18611: 18601: 18600: 18586: 18585: 18564: 18557: 18556: 18546: 18542: 18529: 18525: 18524: 18523: 18517: 18515: 18514: 18510: 18506: 18505: 18484: 18474: 18473: 18458: 18457: 18428: 18421: 18420: 18411: 18407: 18385: 18384: 18375: 18371: 18370: 18369: 18363: 18361: 18354: 18353: 18334: 18309: 18305: 18304: 18303: 18297: 18295: 18284: 18263: 18256: 18255: 18243: 18242: 18208: 18207: 18169: 18167: 18166: 18161: 18159: 18158: 18149: 18148: 18133: 18132: 18110: 18108: 18107: 18102: 18008:This applies to 17997: 17995: 17994: 17989: 17983: 17978: 17951: 17950: 17938: 17937: 17913: 17912: 17900: 17899: 17887: 17883: 17882: 17881: 17875: 17873: 17866: 17865: 17846: 17835: 17830: 17793: 17791: 17790: 17785: 17782: 17777: 17768: 17767: 17749: 17748: 17739: 17738: 17726: 17725: 17707: 17706: 17697: 17696: 17684: 17683: 17668: 17667: 17655: 17654: 17642: 17641: 17620: 17618: 17617: 17612: 17609: 17604: 17595: 17594: 17579: 17578: 17569: 17568: 17552: 17550: 17549: 17544: 17541: 17536: 17527: 17526: 17511: 17510: 17501: 17500: 17454: 17452: 17451: 17446: 17438: 17434: 17433: 17432: 17418: 17414: 17413: 17386: 17384: 17383: 17378: 17370: 17369: 17357: 17356: 17347: 17346: 17304:slice, we get a 17284: 17282: 17281: 17276: 17271: 17270: 17246: 17244: 17237: 17236: 17220: 17219: 17218: 17208: 17201: 17200: 17181: 17179: 17178: 17173: 17171: 17170: 17155:for the current 17151:But this is the 17147: 17145: 17144: 17139: 17131: 17127: 17126: 17125: 17101: 17099: 17092: 17091: 17075: 17074: 17073: 17063: 17056: 17055: 17029: 17027: 17026: 17021: 17019: 17012: 17011: 17002: 16996: 16995: 16986: 16985: 16967: 16963: 16962: 16953: 16935: 16933: 16926: 16925: 16909: 16908: 16907: 16897: 16895: 16894: 16876: 16875: 16870: 16851: 16847: 16846: 16844: 16837: 16836: 16820: 16819: 16818: 16808: 16806: 16805: 16793: 16791: 16783: 16782: 16781: 16771: 16764: 16763: 16747: 16743: 16739: 16738: 16733: 16726: 16725: 16719: 16718: 16635: 16633: 16632: 16627: 16601: 16600: 16573: 16572: 16554: 16553: 16530: 16528: 16527: 16522: 16511: 16510: 16501: 16500: 16476: 16475: 16446: 16444: 16443: 16438: 16436: 16435: 16377: 16376: 16367: 16366: 16351: 16347: 16343: 16342: 16337: 16330: 16329: 16323: 16322: 16284: 16282: 16281: 16276: 16274: 16273: 16250:solutions. (See 16245: 16243: 16242: 16237: 16222: 16220: 16219: 16214: 16206: 16204: 16187: 16177: 16176: 16166: 16154: 16152: 16151: 16146: 16134: 16132: 16131: 16126: 16124: 16123: 16106: 16104: 16103: 16098: 16086: 16084: 16083: 16078: 16076: 16075: 16054: 16052: 16051: 16046: 16022: 16020: 16019: 16014: 15995: 15993: 15992: 15987: 15979: 15978: 15947: 15946: 15919: 15918: 15912: 15911: 15888: 15887: 15872: 15870: 15869: 15864: 15862: 15861: 15848: 15846: 15845: 15840: 15824: 15822: 15821: 15816: 15797: 15795: 15794: 15789: 15775: 15774: 15756: 15755: 15731: 15729: 15728: 15723: 15712: 15711: 15691: 15689: 15688: 15683: 15681: 15669: 15667: 15666: 15661: 15659: 15640: 15638: 15637: 15632: 15627: 15619: 15618: 15609: 15608: 15585: 15583: 15582: 15577: 15575: 15574: 15569: 15552: 15550: 15549: 15544: 15542: 15541: 15536: 15523: 15521: 15520: 15515: 15513: 15512: 15494: 15493: 15446:cotangent bundle 15443: 15441: 15440: 15435: 15433: 15381:smooth functions 15374: 15372: 15371: 15366: 15364: 15363: 15326: 15324: 15323: 15318: 15312: 15311: 15302: 15298: 15296: 15295: 15294: 15286: 15285: 15284: 15269: 15261: 15252: 15248: 15247: 15246: 15230: 15229: 15220: 15219: 15214: 15213: 15206: 15204: 15203: 15202: 15194: 15193: 15192: 15177: 15169: 15159: 15158: 15136: 15134: 15133: 15128: 15120: 15119: 15110: 15108: 15107: 15106: 15090: 15067: 15065: 15064: 15059: 15053: 15052: 15043: 15042: 15037: 15036: 15023: 15022: 15007: 15006: 14997: 14996: 14988: 14972: 14970: 14969: 14964: 14958: 14957: 14948: 14946: 14945: 14944: 14931: 14930: 14929: 14916: 14911: 14910: 14901: 14900: 14895: 14894: 14877: 14875: 14874: 14869: 14861: 14860: 14852: 14851: 14850: 14832: 14830: 14829: 14824: 14822: 14821: 14806:direction. When 14801: 14799: 14798: 14793: 14791: 14790: 14770: 14768: 14767: 14762: 14760: 14759: 14750: 14749: 14744: 14743: 14726: 14724: 14723: 14718: 14716: 14712: 14711: 14702: 14701: 14696: 14695: 14682: 14681: 14672: 14671: 14663: 14657: 14656: 14637: 14636: 14620: 14619: 14600: 14599: 14566: 14564: 14563: 14558: 14546: 14542: 14541: 14540: 14528: 14524: 14523: 14522: 14510: 14509: 14501: 14500: 14499: 14485: 14484: 14464: 14463: 14454: 14453: 14445: 14442: 14440: 14439: 14438: 14430: 14429: 14428: 14413: 14405: 14398: 14396: 14395: 14394: 14378: 14363: 14361: 14360: 14355: 14340: 14339: 14330: 14326: 14325: 14324: 14312: 14308: 14307: 14306: 14294: 14293: 14285: 14284: 14283: 14269: 14268: 14248: 14247: 14238: 14237: 14229: 14226: 14224: 14223: 14222: 14214: 14213: 14212: 14197: 14189: 14182: 14180: 14179: 14178: 14162: 14160: 14159: 14137: 14135: 14134: 14129: 14127: 14120: 14116: 14115: 14114: 14105: 14104: 14096: 14093: 14091: 14090: 14089: 14081: 14080: 14079: 14064: 14056: 14049: 14047: 14046: 14045: 14029: 14024: 14023: 14015: 14014: 14013: 14002: 14001: 13993: 13990: 13988: 13987: 13986: 13978: 13977: 13976: 13961: 13953: 13948: 13947: 13938: 13937: 13929: 13926: 13922: 13920: 13919: 13918: 13910: 13909: 13908: 13893: 13885: 13879: 13877: 13876: 13875: 13859: 13855: 13846: 13842: 13841: 13837: 13836: 13835: 13823: 13822: 13814: 13813: 13812: 13798: 13797: 13777: 13773: 13772: 13771: 13759: 13758: 13750: 13749: 13748: 13734: 13733: 13710: 13693: 13691: 13690: 13685: 13683: 13681: 13680: 13679: 13666: 13658: 13653: 13649: 13647: 13646: 13645: 13637: 13636: 13635: 13620: 13612: 13606: 13604: 13603: 13602: 13586: 13571: 13569: 13568: 13563: 13557: 13553: 13552: 13551: 13542: 13541: 13533: 13525: 13523: 13522: 13521: 13505: 13500: 13498: 13497: 13496: 13483: 13482: 13481: 13468: 13466: 13465: 13457: 13451: 13450: 13442: 13441: 13440: 13429: 13428: 13420: 13401: 13399: 13398: 13393: 13387: 13386: 13378: 13377: 13376: 13365: 13364: 13356: 13353: 13351: 13350: 13349: 13341: 13340: 13339: 13324: 13316: 13311: 13310: 13301: 13300: 13292: 13289: 13287: 13286: 13285: 13272: 13264: 13259: 13255: 13254: 13250: 13249: 13248: 13236: 13235: 13227: 13226: 13225: 13211: 13210: 13190: 13186: 13185: 13184: 13172: 13171: 13163: 13162: 13161: 13147: 13146: 13111: 13109: 13108: 13103: 13088: 13087: 13078: 13074: 13073: 13069: 13068: 13067: 13055: 13051: 13050: 13049: 13037: 13036: 13028: 13027: 13026: 13012: 13011: 12989: 12987: 12986: 12985: 12969: 12964: 12960: 12959: 12955: 12954: 12953: 12941: 12940: 12932: 12931: 12930: 12916: 12915: 12895: 12891: 12890: 12889: 12877: 12876: 12868: 12867: 12866: 12852: 12851: 12824: 12823: 12797: 12795: 12794: 12789: 12783: 12781: 12780: 12779: 12766: 12765: 12764: 12751: 12746: 12745: 12737: 12736: 12735: 12711: 12709: 12708: 12703: 12692: 12691: 12682: 12678: 12677: 12676: 12664: 12663: 12655: 12654: 12653: 12639: 12638: 12621: 12620: 12605: 12604: 12595: 12591: 12590: 12589: 12577: 12576: 12568: 12567: 12566: 12552: 12551: 12534: 12533: 12532: 12531: 12496: 12494: 12493: 12488: 12482: 12478: 12477: 12464: 12463: 12454: 12453: 12445: 12439: 12435: 12434: 12421: 12420: 12408: 12404: 12403: 12390: 12389: 12362: 12360: 12359: 12354: 12352: 12351: 12329: 12327: 12326: 12321: 12315: 12311: 12310: 12297: 12296: 12281: 12277: 12276: 12263: 12262: 12250: 12246: 12245: 12232: 12231: 12219: 12218: 12196: 12194: 12193: 12188: 12186: 12185: 12170: 12169: 12157: 12156: 12144: 12143: 12113:where the index 12105:where the index 12104: 12102: 12101: 12096: 12094: 12093: 12066: 12064: 12063: 12058: 12053: 12051: 12043: 12035: 12033: 12031: 12030: 12029: 12024: 12019: 12012: 12004: 11996: 11992: 11985: 11984: 11979: 11974: 11971: 11969: 11968: 11967: 11962: 11957: 11950: 11942: 11925: 11923: 11922: 11917: 11909: 11907: 11906: 11897: 11889: 11873: 11871: 11870: 11865: 11863: 11861: 11853: 11845: 11843: 11841: 11840: 11839: 11834: 11829: 11822: 11814: 11806: 11802: 11795: 11794: 11789: 11784: 11781: 11779: 11778: 11769: 11761: 11759: 11757: 11756: 11755: 11750: 11745: 11738: 11730: 11710: 11708: 11707: 11702: 11700: 11690: 11689: 11677: 11675: 11667: 11659: 11657: 11655: 11654: 11653: 11648: 11643: 11636: 11628: 11620: 11619: 11607: 11605: 11597: 11589: 11587: 11585: 11584: 11583: 11578: 11573: 11566: 11558: 11553: 11548: 11538: 11537: 11525: 11524: 11519: 11514: 11511: 11509: 11508: 11499: 11491: 11489: 11487: 11486: 11485: 11480: 11475: 11468: 11460: 11449: 11448: 11436: 11435: 11430: 11425: 11422: 11420: 11419: 11410: 11402: 11400: 11398: 11397: 11396: 11391: 11386: 11379: 11371: 11360: 11359: 11344: 11343: 11331: 11330: 11325: 11320: 11311: 11310: 11298: 11278: 11277: 11262: 11261: 11249: 11248: 11243: 11238: 11229: 11228: 11216: 11203: 11178: 11176: 11175: 11170: 11165: 11164: 11159: 11154: 11151: 11149: 11148: 11133: 11129: 11128: 11118: 11116: 11114: 11113: 11112: 11107: 11102: 11095: 11087: 11082: 11080: 11072: 11064: 11062: 11060: 11059: 11050: 11042: 11037: 11036: 11031: 11026: 11023: 11021: 11020: 11005: 11001: 11000: 10990: 10988: 10986: 10985: 10984: 10979: 10974: 10967: 10959: 10954: 10952: 10944: 10936: 10934: 10930: 10929: 10927: 10926: 10925: 10920: 10915: 10908: 10900: 10898: 10896: 10885: 10875: 10871: 10870: 10868: 10860: 10852: 10850: 10848: 10847: 10846: 10841: 10836: 10829: 10821: 10814: 10812: 10801: 10786: 10784: 10783: 10778: 10776: 10762: 10761: 10756: 10751: 10748: 10746: 10745: 10730: 10726: 10725: 10715: 10713: 10711: 10710: 10709: 10704: 10699: 10692: 10684: 10679: 10677: 10669: 10661: 10659: 10657: 10656: 10647: 10639: 10636: 10635: 10634: 10624: 10623: 10622: 10605: 10600: 10590: 10589: 10577: 10576: 10571: 10566: 10563: 10561: 10560: 10551: 10543: 10541: 10539: 10538: 10537: 10532: 10527: 10520: 10512: 10501: 10500: 10488: 10487: 10482: 10477: 10474: 10472: 10471: 10462: 10454: 10452: 10450: 10449: 10448: 10443: 10438: 10431: 10423: 10412: 10411: 10396: 10395: 10383: 10382: 10377: 10372: 10363: 10362: 10350: 10330: 10329: 10314: 10313: 10301: 10300: 10295: 10290: 10281: 10280: 10268: 10255: 10241: 10239: 10231: 10230: 10218: 10193: 10191: 10190: 10185: 10183: 10172: 10171: 10166: 10161: 10158: 10156: 10155: 10146: 10138: 10136: 10134: 10133: 10132: 10127: 10122: 10115: 10107: 10099: 10098: 10093: 10088: 10085: 10081: 10080: 10079: 10074: 10069: 10066: 10064: 10063: 10062: 10053: 10041: 10037: 10036: 10026: 10019: 10017: 10016: 10015: 10010: 10005: 9998: 9990: 9982: 9981: 9976: 9971: 9968: 9966: 9965: 9956: 9948: 9946: 9944: 9943: 9934: 9926: 9918: 9910: 9909: 9904: 9899: 9896: 9894: 9893: 9884: 9876: 9874: 9872: 9871: 9870: 9865: 9860: 9853: 9845: 9837: 9836: 9831: 9826: 9823: 9819: 9818: 9816: 9815: 9806: 9798: 9796: 9794: 9783: 9776: 9774: 9773: 9772: 9767: 9762: 9755: 9747: 9739: 9738: 9733: 9728: 9725: 9723: 9722: 9713: 9705: 9703: 9699: 9698: 9696: 9695: 9694: 9689: 9684: 9677: 9669: 9667: 9665: 9654: 9640: 9636: 9632: 9631: 9626: 9621: 9618: 9616: 9615: 9606: 9598: 9596: 9594: 9593: 9592: 9587: 9582: 9575: 9567: 9560: 9558: 9547: 9528: 9526: 9525: 9520: 9518: 9504: 9500: 9496: 9495: 9490: 9485: 9482: 9480: 9479: 9470: 9462: 9457: 9456: 9451: 9446: 9443: 9441: 9440: 9425: 9421: 9420: 9410: 9402: 9401: 9396: 9395: 9390: 9385: 9381: 9379: 9378: 9377: 9368: 9356: 9352: 9351: 9341: 9331: 9329: 9328: 9327: 9322: 9317: 9310: 9302: 9297: 9293: 9292: 9290: 9282: 9274: 9266: 9265: 9260: 9255: 9252: 9250: 9249: 9240: 9232: 9222: 9220: 9219: 9210: 9202: 9199: 9198: 9197: 9187: 9186: 9185: 9168: 9163: 9153: 9152: 9137: 9136: 9124: 9123: 9118: 9113: 9104: 9103: 9091: 9071: 9070: 9055: 9054: 9042: 9041: 9036: 9031: 9022: 9021: 9009: 8996: 8982: 8980: 8972: 8971: 8959: 8922: 8920: 8919: 8914: 8912: 8908: 8893: 8870: 8859: 8858: 8853: 8848: 8824: 8813: 8805: 8803: 8802: 8793: 8785: 8759: 8748: 8730: 8720: 8719: 8709: 8699: 8698: 8678: 8674: 8659: 8645: 8634: 8630: 8629: 8624: 8619: 8609: 8598: 8594: 8581: 8571: 8570: 8560: 8550: 8549: 8519: 8494: 8492: 8491: 8486: 8469: 8458: 8457: 8452: 8447: 8423: 8412: 8404: 8402: 8401: 8392: 8384: 8361: 8350: 8348: 8337: 8329: 8318: 8314: 8313: 8308: 8303: 8287: 8286: 8281: 8276: 8252: 8250: 8249: 8244: 8242: 8217: 8206: 8174: 8157: 8146: 8142: 8120: 8096: 8053: 8051: 8050: 8045: 8031: 8029: 8028: 8019: 8011: 7997: 7995: 7994: 7993: 7988: 7983: 7976: 7968: 7966: 7964: 7953: 7936: 7934: 7933: 7928: 7901: 7900: 7895: 7890: 7875: 7863: 7862: 7861: 7851: 7850: 7849: 7802:gauge symmetries 7792: 7790: 7789: 7784: 7777: 7776: 7775: 7763: 7759: 7755: 7753: 7752: 7751: 7738: 7737: 7736: 7723: 7718: 7717: 7708: 7706: 7705: 7704: 7691: 7683: 7670: 7669: 7643: 7641: 7640: 7635: 7630: 7629: 7617: 7616: 7604: 7603: 7591: 7590: 7581: 7580: 7568: 7567: 7499: 7497: 7496: 7491: 7484: 7483: 7482: 7473: 7472: 7454: 7453: 7442: 7436: 7432: 7431: 7333: 7331: 7330: 7325: 7313: 7311: 7310: 7305: 7293: 7291: 7290: 7285: 7270: 7268: 7267: 7262: 7254: 7253: 7245: 7242: 7241: 7236: 7233: 7232: 7210: 7208: 7207: 7202: 7200: 7199: 7192: 7187: 7175: 7174: 7159: 7155: 7153: 7152: 7151: 7135: 7134: 7133: 7123: 7114: 7112: 7111: 7110: 7094: 7093: 7092: 7082: 7077: 7076: 7066: 7061: 7049: 7048: 7041: 7036: 7021: 7020: 7015: 7012: 7011: 6992: 6990: 6989: 6984: 6973:) into a tensor 6972: 6970: 6969: 6964: 6952: 6950: 6949: 6944: 6932: 6930: 6929: 6924: 6897: 6895: 6894: 6889: 6887: 6886: 6879: 6874: 6858: 6853: 6834: 6832: 6831: 6826: 6824: 6820: 6818: 6813: 6804: 6803: 6791: 6790: 6776: 6775: 6766: 6762: 6761: 6748: 6747: 6731: 6729: 6728: 6723: 6715: 6714: 6704: 6699: 6684: 6683: 6664: 6662: 6661: 6656: 6654: 6653: 6637: 6635: 6634: 6629: 6627: 6626: 6611: 6610: 6594: 6592: 6591: 6586: 6584: 6583: 6567: 6565: 6564: 6559: 6557: 6553: 6551: 6546: 6537: 6536: 6524: 6523: 6500: 6499: 6473: 6471: 6470: 6465: 6453: 6451: 6450: 6445: 6442: 6437: 6428: 6427: 6415: 6414: 6402: 6401: 6381: 6379: 6378: 6373: 6371: 6367: 6366: 6365: 6353: 6348: 6347: 6335: 6305: 6303: 6302: 6297: 6286: 6285: 6276: 6275: 6255:conservation law 6249: 6247: 6246: 6241: 6229: 6227: 6226: 6221: 6198: 6196: 6195: 6190: 6188: 6187: 6175: 6173: 6172: 6171: 6155: 6154: 6153: 6143: 6137: 6132: 6119: 6114: 6092: 6090: 6089: 6084: 6069: 6067: 6066: 6061: 6055: 6050: 6041: 6040: 6031: 6030: 6018: 6017: 6008: 6007: 5988: 5986: 5985: 5980: 5975: 5974: 5965: 5964: 5933: 5931: 5930: 5925: 5879: 5877: 5876: 5871: 5866: 5865: 5856: 5855: 5840: 5839: 5830: 5829: 5813: 5811: 5810: 5805: 5803: 5802: 5789: 5787: 5786: 5781: 5779: 5778: 5765: 5763: 5762: 5757: 5745: 5743: 5742: 5737: 5725: 5723: 5722: 5717: 5715: 5714: 5698: 5696: 5695: 5690: 5675: 5673: 5672: 5667: 5634: 5632: 5631: 5626: 5600: 5598: 5597: 5592: 5587: 5586: 5576: 5572: 5571: 5570: 5555: 5554: 5534: 5533: 5521: 5520: 5500: 5498: 5497: 5492: 5481: 5463: 5461: 5460: 5455: 5419:angular momentum 5397: 5395: 5394: 5389: 5384: 5376: 5368: 5364: 5363: 5355: 5342: 5334: 5330: 5329: 5321: 5308: 5300: 5292: 5290: 5289: 5288: 5283: 5278: 5271: 5263: 5248: 5246: 5245: 5240: 5235: 5227: 5219: 5173: 5171: 5170: 5165: 5160: 5152: 5137: 5129: 5115:by the equation 5089:angular momentum 5050: 5048: 5047: 5042: 5037: 5035: 5034: 5033: 5028: 5027: 5018: 5011: 5003: 4998: 4997: 4955: = 1, 4951:. In that case, 4906: 4904: 4903: 4898: 4887: 4886: 4881: 4876: 4870: 4868: 4867: 4866: 4861: 4856: 4849: 4841: 4810: = 1, 4806:. In this case, 4769: 4767: 4766: 4761: 4759: 4758: 4753: 4744: 4742: 4741: 4740: 4735: 4730: 4723: 4715: 4710: 4709: 4700: 4696: 4689: 4688: 4683: 4678: 4672: 4670: 4669: 4668: 4663: 4658: 4651: 4643: 4623:showed that the 4543: 4541: 4540: 4535: 4533: 4524: 4523: 4522: 4517: 4511: 4510: 4500: 4484: 4472: 4471: 4462: 4461: 4451: 4382: 4380: 4379: 4374: 4372: 4363: 4362: 4351: 4343: 4342: 4337: 4324: 4300: 4299: 4267:. One may write 4174: 4172: 4171: 4166: 4164: 4162: 4161: 4160: 4155: 4154: 4146: 4138: 4130: 4125: 4124: 4102: 4100: 4099: 4094: 4087: 4080: 4078: 4070: 4069: 4068: 4055: 4050: 4046: 4044: 4043: 4042: 4037: 4036: 4028: 4020: 4012: 4006: 4004: 3993: 3971: 3969: 3968: 3963: 3956: 3955: 3953: 3952: 3943: 3935: 3930: 3926: 3924: 3923: 3922: 3917: 3912: 3905: 3897: 3891: 3889: 3878: 3844: 3842: 3841: 3836: 3829: 3828: 3826: 3818: 3817: 3808: 3803: 3802: 3797: 3792: 3768: 3766: 3765: 3760: 3753: 3736: 3735: 3730: 3725: 3719: 3613:on the basis of 3581: 3579: 3578: 3573: 3566: 3559: 3558: 3550: 3544: 3542: 3534: 3526: 3511:with respect to 3493:conservation law 3483:conservation law 3466: 3464: 3463: 3458: 3443: 3441: 3440: 3435: 3433: 3432: 3416: 3414: 3413: 3408: 3393: 3391: 3390: 3385: 3367: 3365: 3364: 3359: 3357: 3356: 3337: 3335: 3334: 3329: 3311: 3309: 3308: 3303: 3288: 3286: 3285: 3280: 3278: 3277: 3261: 3259: 3258: 3253: 3251: 3250: 3231: 3229: 3228: 3223: 3212: 3211: 3192: 3190: 3189: 3184: 3172: 3170: 3169: 3166:{\displaystyle } 3164: 3159: 3158: 3146: 3145: 3123: 3121: 3120: 3115: 3104: 3103: 3081: 3079: 3078: 3073: 3049: 3047: 3046: 3041: 3039: 3035: 3034: 3033: 3028: 3027: 3019: 3015: 3011: 3010: 3009: 3004: 3003: 2995: 2988: 2971: 2939: 2937: 2936: 2931: 2926: 2922: 2921: 2920: 2915: 2914: 2906: 2902: 2900: 2899: 2898: 2893: 2892: 2884: 2876: 2868: 2846: 2842: 2841: 2840: 2835: 2834: 2826: 2819: 2817: 2816: 2815: 2810: 2809: 2801: 2793: 2785: 2746: 2744: 2743: 2738: 2723: 2721: 2720: 2715: 2688: 2686: 2685: 2680: 2678: 2677: 2668: 2664: 2663: 2662: 2657: 2656: 2648: 2641: 2624: 2608: 2606: 2605: 2600: 2598: 2597: 2585: 2584: 2572: 2571: 2555: 2553: 2552: 2547: 2545: 2544: 2520: 2518: 2517: 2512: 2504: 2500: 2499: 2498: 2490: 2484: 2461: 2459: 2458: 2453: 2441: 2439: 2438: 2433: 2428: 2424: 2423: 2422: 2414: 2408: 2385: 2383: 2382: 2377: 2369: 2368: 2356: 2352: 2348: 2346: 2345: 2344: 2336: 2329: 2321: 2308: 2307: 2295: 2291: 2287: 2285: 2284: 2283: 2275: 2268: 2260: 2246: 2244: 2243: 2238: 2223: 2221: 2220: 2215: 2213: 2212: 2196: 2194: 2193: 2188: 2186: 2185: 2167: 2165: 2164: 2159: 2151: 2149: 2148: 2147: 2139: 2132: 2124: 2110: 2108: 2107: 2106: 2098: 2091: 2083: 2076: 2072: 2068: 2067: 2062: 2054: 2044: 2042: 2041: 2040: 2032: 2025: 2017: 2009: 2008: 2000: 1994: 1992: 1991: 1990: 1982: 1975: 1967: 1934: 1932: 1931: 1926: 1924: 1923: 1915: 1909: 1908: 1902: 1901: 1893: 1887: 1876: 1875: 1850: 1848: 1847: 1842: 1837: 1823: 1822: 1814: 1808: 1807: 1799: 1789: 1787: 1786: 1781: 1767: 1749: 1747: 1746: 1741: 1729: 1727: 1726: 1721: 1706: 1704: 1703: 1698: 1693: 1675: 1673: 1672: 1667: 1665: 1664: 1656: 1646: 1644: 1643: 1638: 1636: 1635: 1627: 1617: 1615: 1614: 1609: 1597: 1595: 1594: 1589: 1568: 1566: 1565: 1560: 1548: 1546: 1545: 1540: 1528: 1526: 1525: 1520: 1505: 1503: 1502: 1499:{\displaystyle } 1497: 1492: 1491: 1479: 1478: 1459: 1457: 1456: 1451: 1436: 1434: 1433: 1428: 1416: 1398: 1396: 1395: 1390: 1370: 1368: 1367: 1362: 1343: 1341: 1340: 1335: 1311: 1309: 1308: 1303: 1270: 1268: 1267: 1262: 1172:conservation law 1088:angular momentum 1013:conservation law 986: 979: 972: 920: 885: 851: 850: 847: 814:Surface integral 757: 756: 753: 661: 660: 657: 617:Limit comparison 537: 536: 533: 424:Riemann integral 377: 376: 373: 333:L'Hôpital's rule 290:Taylor's theorem 211: 210: 207: 151: 149: 148: 143: 95: 86: 81: 51: 50: 21: 23856: 23855: 23851: 23850: 23849: 23847: 23846: 23845: 23796: 23795: 23733: 23712: 23710: 23706: 23673: 23576: 23574: 23555:physics/9807044 23511:physics/0503066 23483: 23473: 23463:Springer-Verlag 23447: 23437:Springer-Verlag 23414: 23412: 23392: 23382:Springer-Verlag 23363: 23361: 23334: 23302: 23300:Further reading 23297: 23281: 23280: 23224:Wald, Robert M. 23220: 23216: 23209: 23193: 23189: 23182: 23163: 23154: 23147: 23130: 23121: 23114: 23090: 23086: 23079: 23055: 23051: 23042: 23038: 23031: 23015: 23011: 22968: 22964: 22949: 22935: 22931: 22916: 22900: 22896: 22881: 22865: 22858: 22839: 22835: 22831: 22826: 22817:Goldstone boson 22781: 22774: 22767: 22760: 22757: 22718:electric charge 22676:linear momentum 22663: 22624: 22620: 22614: 22610: 22608: 22605: 22604: 22587: 22586: 22572: 22568: 22553: 22549: 22540: 22536: 22526: 22525: 22521: 22515: 22511: 22488: 22484: 22478: 22474: 22473: 22469: 22460: 22456: 22447: 22446: 22440: 22436: 22410: 22409: 22394: 22390: 22383: 22378: 22377: 22373: 22366: 22360: 22356: 22352: 22350: 22347: 22346: 22320: 22319: 22313: 22309: 22300: 22296: 22294: 22291: 22290: 22262: 22261: 22255: 22251: 22250: 22246: 22240: 22236: 22222: 22218: 22212: 22208: 22193: 22189: 22180: 22176: 22170: 22166: 22156: 22155: 22151: 22145: 22141: 22139: 22136: 22135: 22095: 22091: 22085: 22081: 22080: 22076: 22070: 22066: 22043: 22039: 22027: 22023: 22017: 22013: 22007: 22003: 21991: 21987: 21986: 21982: 21973: 21969: 21957: 21956: 21948: 21945: 21944: 21925: 21922: 21921: 21871: 21867: 21861: 21857: 21831: 21828: 21827: 21803: 21802: 21793: 21789: 21778: 21774: 21759: 21755: 21746: 21742: 21732: 21731: 21727: 21715: 21714: 21705: 21701: 21678: 21674: 21658: 21654: 21648: 21647: 21637: 21622: 21621: 21617: 21615: 21612: 21611: 21593: 21562: 21546: 21544: 21543: 21542: 21525: 21524: 21505: 21493: 21492: 21488: 21486: 21484: 21481: 21480: 21457: 21446: 21445: 21444: 21442: 21439: 21438: 21412: 21411: 21409: 21406: 21405: 21388: 21387: 21378: 21367: 21366: 21365: 21345: 21344: 21335: 21334: 21323: 21312: 21311: 21310: 21304: 21300: 21288: 21272: 21270: 21269: 21268: 21262: 21258: 21257: 21253: 21247: 21234: 21233: 21222: 21221: 21208: 21197: 21196: 21195: 21191: 21180: 21179: 21165: 21164: 21155: 21139: 21137: 21136: 21135: 21131: 21126: 21125: 21121: 21115: 21104: 21093: 21092: 21088: 21086: 21083: 21082: 21056: 21045: 21044: 21043: 21037: 21033: 21027: 21009: 21008: 21006: 21003: 21002: 20982: 20977: 20967: 20963: 20957: 20943: 20938: 20936: 20933: 20932: 20915: 20914: 20905: 20900: 20889: 20888: 20881: 20877: 20871: 20858: 20857: 20846: 20835: 20834: 20833: 20824: 20813: 20812: 20811: 20810: 20806: 20797: 20793: 20787: 20783: 20768: 20755: 20750: 20739: 20738: 20731: 20727: 20721: 20710: 20701: 20700: 20691: 20687: 20683: 20681: 20678: 20677: 20651: 20646: 20616: 20611: 20606: 20602: 20596: 20592: 20590: 20587: 20586: 20561: 20560: 20558: 20555: 20554: 20535: 20532: 20531: 20514: 20513: 20488: 20477: 20476: 20475: 20466: 20455: 20454: 20453: 20452: 20448: 20439: 20435: 20423: 20410: 20400: 20384: 20382: 20381: 20380: 20376: 20375: 20364: 20360: 20358: 20352: 20341: 20336: 20332: 20320: 20319: 20283: 20281: 20280: 20257: 20256: 20255: 20251: 20245: 20244: 20234: 20219: 20218: 20214: 20208: 20207: 20203: 20201: 20198: 20197: 20191: 20150: 20146: 20144: 20143: 20134: 20123: 20122: 20121: 20117: 20109: 20107: 20101: 20090: 20084: 20081: 20080: 20046: 20038: 20035: 20034: 20017: 20016: 19992: 19987: 19976: 19975: 19968: 19954: 19945: 19944: 19918: 19913: 19902: 19901: 19894: 19880: 19879: 19875: 19866: 19861: 19850: 19849: 19842: 19826: 19825: 19810: 19806: 19791: 19780: 19779: 19778: 19774: 19766: 19764: 19758: 19747: 19736: 19729: 19727: 19724: 19723: 19682: 19677: 19666: 19665: 19658: 19644: 19636: 19633: 19632: 19609: 19598: 19597: 19596: 19587: 19583: 19579: 19562: 19560: 19554: 19541: 19530: 19529: 19528: 19522: 19511: 19510: 19509: 19503: 19467: 19462: 19451: 19450: 19443: 19429: 19428: 19424: 19414: 19409: 19392: 19389: 19388: 19346: 19345: 19319: 19316: 19315: 19275: 19274: 19229: 19224: 19213: 19212: 19205: 19194: 19180: 19179: 19175: 19163: 19162: 19130: 19129: 19113: 19109: 19096: 19081: 19080: 19076: 19074: 19071: 19070: 19040: 19035: 18973: 18972: 18957: 18953: 18947: 18943: 18936: 18931: 18930: 18926: 18925: 18921: 18915: 18911: 18890: 18886: 18875: 18874: 18859: 18855: 18849: 18845: 18838: 18833: 18832: 18828: 18799: 18798: 18783: 18779: 18772: 18767: 18766: 18762: 18753: 18749: 18748: 18744: 18738: 18734: 18732: 18729: 18728: 18706: 18703: 18702: 18685: 18684: 18675: 18671: 18642: 18641: 18626: 18622: 18615: 18610: 18609: 18605: 18596: 18592: 18591: 18587: 18581: 18577: 18562: 18561: 18552: 18548: 18519: 18518: 18501: 18497: 18496: 18492: 18488: 18483: 18482: 18478: 18469: 18465: 18453: 18452: 18442: 18438: 18426: 18425: 18416: 18412: 18380: 18376: 18365: 18364: 18349: 18345: 18338: 18333: 18332: 18328: 18299: 18298: 18288: 18283: 18282: 18278: 18277: 18273: 18261: 18260: 18251: 18247: 18238: 18237: 18212: 18203: 18202: 18183: 18181: 18178: 18177: 18154: 18150: 18144: 18140: 18128: 18127: 18119: 18116: 18115: 18096: 18093: 18092: 18006: 17979: 17974: 17946: 17942: 17933: 17929: 17908: 17904: 17895: 17891: 17877: 17876: 17861: 17857: 17850: 17845: 17844: 17840: 17831: 17826: 17820: 17817: 17816: 17814: 17807: 17800: 17778: 17773: 17763: 17759: 17744: 17743: 17734: 17730: 17721: 17717: 17702: 17701: 17692: 17688: 17679: 17675: 17663: 17662: 17650: 17646: 17637: 17633: 17628: 17625: 17624: 17605: 17600: 17590: 17586: 17574: 17573: 17564: 17560: 17558: 17555: 17554: 17537: 17532: 17522: 17518: 17506: 17505: 17496: 17492: 17490: 17487: 17486: 17484: 17477: 17469: 17428: 17427: 17409: 17405: 17401: 17397: 17395: 17392: 17391: 17365: 17361: 17352: 17348: 17339: 17335: 17333: 17330: 17329: 17318: 17290:Noether current 17266: 17262: 17232: 17228: 17221: 17214: 17213: 17209: 17207: 17196: 17192: 17190: 17187: 17186: 17166: 17162: 17160: 17157: 17156: 17121: 17117: 17087: 17083: 17076: 17069: 17068: 17064: 17062: 17061: 17057: 17051: 17047: 17045: 17042: 17041: 17017: 17016: 17007: 17003: 16998: 16991: 16987: 16978: 16974: 16965: 16964: 16958: 16954: 16949: 16921: 16917: 16910: 16903: 16902: 16898: 16896: 16887: 16883: 16871: 16866: 16865: 16832: 16828: 16821: 16814: 16813: 16809: 16807: 16801: 16797: 16784: 16777: 16776: 16772: 16770: 16769: 16765: 16759: 16755: 16748: 16734: 16729: 16728: 16721: 16720: 16714: 16710: 16709: 16705: 16698: 16696: 16693: 16692: 16596: 16592: 16568: 16567: 16549: 16548: 16546: 16543: 16542: 16538:, where we set 16506: 16502: 16496: 16492: 16471: 16470: 16462: 16459: 16458: 16431: 16427: 16372: 16368: 16359: 16355: 16338: 16333: 16332: 16325: 16324: 16318: 16314: 16313: 16309: 16304: 16301: 16300: 16269: 16268: 16266: 16263: 16262: 16231: 16228: 16227: 16188: 16172: 16171: 16167: 16165: 16163: 16160: 16159: 16140: 16137: 16136: 16119: 16118: 16116: 16113: 16112: 16092: 16089: 16088: 16071: 16070: 16068: 16065: 16064: 16040: 16037: 16036: 16008: 16005: 16004: 15974: 15970: 15942: 15938: 15914: 15913: 15907: 15903: 15883: 15882: 15880: 15877: 15876: 15857: 15856: 15854: 15851: 15850: 15834: 15831: 15830: 15810: 15807: 15806: 15805:, depending on 15770: 15766: 15751: 15750: 15748: 15745: 15744: 15707: 15706: 15704: 15701: 15700: 15677: 15675: 15672: 15671: 15655: 15653: 15650: 15649: 15623: 15614: 15613: 15604: 15603: 15601: 15598: 15597: 15570: 15565: 15564: 15562: 15559: 15558: 15537: 15532: 15531: 15529: 15526: 15525: 15508: 15504: 15489: 15485: 15483: 15480: 15479: 15429: 15427: 15424: 15423: 15359: 15358: 15356: 15353: 15352: 15333: 15307: 15303: 15287: 15280: 15276: 15275: 15274: 15270: 15262: 15260: 15256: 15242: 15238: 15225: 15221: 15215: 15209: 15208: 15207: 15195: 15188: 15184: 15183: 15182: 15178: 15170: 15168: 15167: 15163: 15154: 15150: 15148: 15145: 15144: 15115: 15111: 15102: 15098: 15094: 15089: 15087: 15084: 15083: 15048: 15044: 15038: 15032: 15031: 15030: 15018: 15014: 15002: 14998: 14987: 14986: 14984: 14981: 14980: 14953: 14949: 14940: 14936: 14932: 14925: 14921: 14917: 14915: 14906: 14902: 14896: 14890: 14889: 14888: 14886: 14883: 14882: 14853: 14846: 14842: 14841: 14840: 14838: 14835: 14834: 14833:is a scalar or 14817: 14813: 14811: 14808: 14807: 14786: 14782: 14780: 14777: 14776: 14755: 14751: 14745: 14739: 14738: 14737: 14735: 14732: 14731: 14714: 14713: 14707: 14703: 14697: 14691: 14690: 14689: 14677: 14673: 14662: 14661: 14652: 14648: 14638: 14632: 14628: 14622: 14621: 14615: 14611: 14601: 14595: 14591: 14584: 14582: 14579: 14578: 14536: 14532: 14518: 14514: 14502: 14495: 14491: 14490: 14489: 14480: 14476: 14475: 14471: 14459: 14455: 14444: 14443: 14431: 14424: 14420: 14419: 14418: 14414: 14406: 14404: 14403: 14399: 14390: 14386: 14382: 14377: 14375: 14372: 14371: 14335: 14331: 14320: 14316: 14302: 14298: 14286: 14279: 14275: 14274: 14273: 14264: 14260: 14259: 14255: 14243: 14239: 14228: 14227: 14215: 14208: 14204: 14203: 14202: 14198: 14190: 14188: 14187: 14183: 14174: 14170: 14166: 14161: 14155: 14151: 14149: 14146: 14145: 14125: 14124: 14110: 14106: 14095: 14094: 14082: 14075: 14071: 14070: 14069: 14065: 14057: 14055: 14054: 14050: 14041: 14037: 14033: 14028: 14016: 14009: 14005: 14004: 14003: 13992: 13991: 13979: 13972: 13968: 13967: 13966: 13962: 13954: 13952: 13943: 13939: 13928: 13927: 13911: 13904: 13900: 13899: 13898: 13894: 13886: 13884: 13880: 13871: 13867: 13863: 13858: 13856: 13854: 13848: 13847: 13831: 13827: 13815: 13808: 13804: 13803: 13802: 13793: 13789: 13788: 13784: 13767: 13763: 13751: 13744: 13740: 13739: 13738: 13729: 13725: 13724: 13720: 13716: 13712: 13707: 13705: 13702: 13701: 13675: 13671: 13667: 13659: 13657: 13638: 13631: 13627: 13626: 13625: 13621: 13613: 13611: 13607: 13598: 13594: 13590: 13585: 13583: 13580: 13579: 13547: 13543: 13532: 13531: 13530: 13526: 13517: 13513: 13509: 13504: 13492: 13488: 13484: 13477: 13473: 13469: 13467: 13456: 13455: 13443: 13436: 13432: 13431: 13430: 13419: 13418: 13416: 13413: 13412: 13379: 13372: 13368: 13367: 13366: 13355: 13354: 13342: 13335: 13331: 13330: 13329: 13325: 13317: 13315: 13306: 13302: 13291: 13290: 13281: 13277: 13273: 13265: 13263: 13244: 13240: 13228: 13221: 13217: 13216: 13215: 13206: 13202: 13201: 13197: 13180: 13176: 13164: 13157: 13153: 13152: 13151: 13142: 13138: 13137: 13133: 13129: 13125: 13123: 13120: 13119: 13083: 13079: 13063: 13059: 13045: 13041: 13029: 13022: 13018: 13017: 13016: 13007: 13003: 13002: 12998: 12994: 12990: 12981: 12977: 12973: 12968: 12949: 12945: 12933: 12926: 12922: 12921: 12920: 12911: 12907: 12906: 12902: 12885: 12881: 12869: 12862: 12858: 12857: 12856: 12847: 12843: 12842: 12838: 12834: 12830: 12829: 12825: 12819: 12815: 12813: 12810: 12809: 12775: 12771: 12767: 12760: 12756: 12752: 12750: 12738: 12731: 12727: 12726: 12725: 12723: 12720: 12719: 12687: 12683: 12672: 12668: 12656: 12649: 12645: 12644: 12643: 12634: 12630: 12629: 12625: 12616: 12612: 12600: 12596: 12585: 12581: 12569: 12562: 12558: 12557: 12556: 12547: 12543: 12542: 12538: 12527: 12523: 12522: 12518: 12516: 12513: 12512: 12473: 12469: 12465: 12459: 12455: 12444: 12443: 12430: 12426: 12422: 12416: 12412: 12399: 12395: 12391: 12385: 12381: 12379: 12376: 12375: 12371:may be defined 12347: 12343: 12338: 12335: 12334: 12306: 12302: 12298: 12292: 12288: 12272: 12268: 12264: 12258: 12254: 12241: 12237: 12233: 12227: 12223: 12214: 12210: 12208: 12205: 12204: 12181: 12177: 12165: 12161: 12152: 12148: 12139: 12135: 12133: 12130: 12129: 12089: 12085: 12083: 12080: 12079: 12076: 12044: 12036: 12034: 12020: 12018: 12017: 12013: 12005: 12003: 11975: 11973: 11972: 11958: 11956: 11955: 11951: 11943: 11941: 11940: 11936: 11934: 11931: 11930: 11902: 11898: 11890: 11888: 11886: 11883: 11882: 11854: 11846: 11844: 11830: 11828: 11827: 11823: 11815: 11813: 11785: 11783: 11782: 11774: 11770: 11762: 11760: 11746: 11744: 11743: 11739: 11731: 11729: 11728: 11724: 11722: 11719: 11718: 11698: 11697: 11685: 11681: 11668: 11660: 11658: 11644: 11642: 11641: 11637: 11629: 11627: 11615: 11611: 11598: 11590: 11588: 11574: 11572: 11571: 11567: 11559: 11557: 11552: 11546: 11545: 11533: 11529: 11515: 11513: 11512: 11504: 11500: 11492: 11490: 11476: 11474: 11473: 11469: 11461: 11459: 11444: 11440: 11426: 11424: 11423: 11415: 11411: 11403: 11401: 11387: 11385: 11384: 11380: 11372: 11370: 11355: 11351: 11339: 11335: 11321: 11319: 11318: 11306: 11302: 11294: 11273: 11269: 11257: 11253: 11239: 11237: 11236: 11224: 11220: 11212: 11204: 11202: 11192: 11190: 11187: 11186: 11155: 11153: 11152: 11144: 11134: 11124: 11120: 11119: 11117: 11103: 11101: 11100: 11096: 11088: 11086: 11073: 11065: 11063: 11055: 11051: 11043: 11041: 11027: 11025: 11024: 11016: 11006: 10996: 10992: 10991: 10989: 10975: 10973: 10972: 10968: 10960: 10958: 10945: 10937: 10935: 10916: 10914: 10913: 10909: 10901: 10899: 10889: 10884: 10883: 10879: 10861: 10853: 10851: 10837: 10835: 10834: 10830: 10822: 10820: 10819: 10815: 10805: 10800: 10798: 10795: 10794: 10774: 10773: 10752: 10750: 10749: 10741: 10731: 10721: 10717: 10716: 10714: 10700: 10698: 10697: 10693: 10685: 10683: 10670: 10662: 10660: 10652: 10648: 10640: 10638: 10630: 10626: 10625: 10618: 10614: 10613: 10604: 10598: 10597: 10585: 10581: 10567: 10565: 10564: 10556: 10552: 10544: 10542: 10528: 10526: 10525: 10521: 10513: 10511: 10496: 10492: 10478: 10476: 10475: 10467: 10463: 10455: 10453: 10439: 10437: 10436: 10432: 10424: 10422: 10407: 10403: 10391: 10387: 10373: 10371: 10370: 10358: 10354: 10346: 10325: 10321: 10309: 10305: 10291: 10289: 10288: 10276: 10272: 10264: 10256: 10254: 10232: 10223: 10219: 10217: 10207: 10205: 10202: 10201: 10181: 10180: 10162: 10160: 10159: 10151: 10147: 10139: 10137: 10123: 10121: 10120: 10116: 10108: 10106: 10089: 10087: 10086: 10070: 10068: 10067: 10058: 10054: 10049: 10042: 10032: 10028: 10027: 10025: 10024: 10020: 10006: 10004: 10003: 9999: 9991: 9989: 9972: 9970: 9969: 9961: 9957: 9949: 9947: 9939: 9935: 9927: 9925: 9916: 9915: 9900: 9898: 9897: 9889: 9885: 9877: 9875: 9861: 9859: 9858: 9854: 9846: 9844: 9827: 9825: 9824: 9811: 9807: 9799: 9797: 9787: 9782: 9781: 9777: 9763: 9761: 9760: 9756: 9748: 9746: 9729: 9727: 9726: 9718: 9714: 9706: 9704: 9685: 9683: 9682: 9678: 9670: 9668: 9658: 9653: 9652: 9648: 9641: 9622: 9620: 9619: 9611: 9607: 9599: 9597: 9583: 9581: 9580: 9576: 9568: 9566: 9565: 9561: 9551: 9546: 9542: 9540: 9537: 9536: 9516: 9515: 9486: 9484: 9483: 9475: 9471: 9463: 9461: 9447: 9445: 9444: 9436: 9426: 9416: 9412: 9411: 9409: 9397: 9386: 9384: 9383: 9382: 9373: 9369: 9364: 9357: 9347: 9343: 9342: 9340: 9336: 9332: 9318: 9316: 9315: 9311: 9303: 9301: 9283: 9275: 9273: 9256: 9254: 9253: 9245: 9241: 9233: 9231: 9227: 9223: 9215: 9211: 9203: 9201: 9193: 9189: 9188: 9181: 9177: 9176: 9167: 9161: 9160: 9148: 9144: 9132: 9128: 9114: 9112: 9111: 9099: 9095: 9087: 9066: 9062: 9050: 9046: 9032: 9030: 9029: 9017: 9013: 9005: 8997: 8995: 8973: 8964: 8960: 8958: 8948: 8946: 8943: 8942: 8910: 8909: 8901: 8886: 8863: 8849: 8847: 8846: 8817: 8809: 8798: 8794: 8786: 8784: 8752: 8744: 8715: 8711: 8710: 8694: 8690: 8689: 8676: 8675: 8667: 8652: 8638: 8620: 8618: 8617: 8616: 8602: 8590: 8589: 8566: 8562: 8561: 8545: 8541: 8540: 8529: 8512: 8508: 8506: 8503: 8502: 8462: 8448: 8446: 8445: 8416: 8408: 8397: 8393: 8385: 8383: 8357: 8341: 8336: 8322: 8304: 8302: 8301: 8300: 8277: 8275: 8274: 8272: 8269: 8268: 8240: 8239: 8210: 8202: 8170: 8150: 8138: 8137: 8130: 8116: 8113: 8112: 8089: 8082: 8075: 8073: 8070: 8069: 8024: 8020: 8012: 8010: 7984: 7982: 7981: 7977: 7969: 7967: 7957: 7952: 7950: 7947: 7946: 7891: 7889: 7888: 7871: 7857: 7853: 7852: 7845: 7841: 7840: 7828: 7825: 7824: 7815: 7810: 7768: 7764: 7747: 7743: 7739: 7732: 7728: 7724: 7722: 7713: 7709: 7700: 7696: 7692: 7684: 7682: 7681: 7677: 7665: 7661: 7659: 7656: 7655: 7625: 7621: 7612: 7608: 7596: 7592: 7586: 7582: 7576: 7572: 7563: 7559: 7551: 7548: 7547: 7535:version of the 7478: 7474: 7462: 7458: 7449: 7445: 7424: 7420: 7412: 7409: 7408: 7343:electric charge 7319: 7316: 7315: 7299: 7296: 7295: 7279: 7276: 7275: 7274:(we relabelled 7246: 7244: 7243: 7237: 7235: 7234: 7228: 7224: 7222: 7219: 7218: 7195: 7194: 7188: 7183: 7167: 7163: 7144: 7140: 7136: 7129: 7128: 7124: 7122: 7118: 7103: 7099: 7095: 7088: 7087: 7083: 7081: 7072: 7068: 7062: 7057: 7044: 7043: 7037: 7032: 7016: 7014: 7013: 7007: 7003: 7001: 6998: 6997: 6978: 6975: 6974: 6958: 6955: 6954: 6938: 6935: 6934: 6918: 6915: 6914: 6912: 6882: 6881: 6875: 6870: 6854: 6849: 6843: 6840: 6839: 6814: 6809: 6799: 6795: 6786: 6782: 6781: 6777: 6771: 6770: 6757: 6753: 6749: 6743: 6742: 6740: 6737: 6736: 6710: 6706: 6700: 6695: 6679: 6675: 6673: 6670: 6669: 6649: 6645: 6643: 6640: 6639: 6622: 6618: 6606: 6602: 6600: 6597: 6596: 6579: 6575: 6573: 6570: 6569: 6547: 6542: 6532: 6528: 6519: 6515: 6514: 6510: 6495: 6491: 6483: 6480: 6479: 6476:Kronecker delta 6459: 6456: 6455: 6438: 6433: 6423: 6419: 6410: 6406: 6397: 6393: 6391: 6388: 6387: 6361: 6357: 6349: 6343: 6339: 6331: 6330: 6326: 6321: 6318: 6317: 6311:electric charge 6281: 6277: 6271: 6267: 6265: 6262: 6261: 6235: 6232: 6231: 6215: 6212: 6211: 6183: 6179: 6164: 6160: 6156: 6149: 6148: 6144: 6142: 6133: 6128: 6115: 6110: 6104: 6101: 6100: 6078: 6075: 6074: 6051: 6046: 6036: 6032: 6026: 6022: 6013: 6012: 6003: 6002: 6000: 5997: 5996: 5970: 5966: 5960: 5956: 5942: 5939: 5938: 5892: 5889: 5888: 5861: 5857: 5851: 5847: 5835: 5834: 5825: 5824: 5822: 5819: 5818: 5798: 5797: 5795: 5792: 5791: 5774: 5773: 5771: 5768: 5767: 5751: 5748: 5747: 5731: 5728: 5727: 5710: 5706: 5704: 5701: 5700: 5684: 5681: 5680: 5643: 5640: 5639: 5620: 5617: 5616: 5582: 5578: 5566: 5562: 5550: 5546: 5539: 5535: 5529: 5528: 5516: 5515: 5513: 5510: 5509: 5477: 5469: 5466: 5465: 5449: 5446: 5445: 5427: 5380: 5372: 5359: 5351: 5350: 5346: 5338: 5325: 5317: 5316: 5312: 5304: 5296: 5279: 5277: 5276: 5272: 5264: 5262: 5260: 5257: 5256: 5231: 5223: 5215: 5213: 5210: 5209: 5156: 5148: 5133: 5125: 5123: 5120: 5119: 5023: 5019: 5017: 5016: 5012: 5004: 5002: 4993: 4989: 4987: 4984: 4983: 4979: 4967: 4950: 4941: 4932: 4923: 4877: 4875: 4874: 4857: 4855: 4854: 4850: 4842: 4840: 4832: 4829: 4828: 4783: 4773:are conserved ( 4754: 4749: 4748: 4731: 4729: 4728: 4724: 4716: 4714: 4705: 4701: 4679: 4677: 4676: 4659: 4657: 4656: 4652: 4644: 4642: 4641: 4637: 4635: 4632: 4631: 4603: 4590: 4571: 4555: 4531: 4530: 4518: 4513: 4512: 4506: 4502: 4496: 4485: 4480: 4474: 4473: 4467: 4463: 4457: 4453: 4447: 4436: 4426: 4424: 4421: 4420: 4370: 4369: 4358: 4347: 4338: 4333: 4332: 4325: 4320: 4317: 4316: 4295: 4291: 4284: 4277: 4275: 4272: 4271: 4247: 4242: 4236: 4228:Albert Einstein 4197: 4190: 4156: 4145: 4144: 4143: 4139: 4131: 4129: 4120: 4116: 4114: 4111: 4110: 4071: 4064: 4060: 4056: 4054: 4038: 4027: 4026: 4025: 4021: 4013: 4011: 4007: 3997: 3992: 3990: 3987: 3986: 3980: 3948: 3944: 3936: 3934: 3913: 3911: 3910: 3906: 3898: 3896: 3892: 3882: 3877: 3875: 3872: 3871: 3819: 3813: 3809: 3807: 3793: 3791: 3790: 3788: 3785: 3784: 3726: 3724: 3723: 3715: 3698: 3695: 3694: 3625:. According to 3549: 3548: 3535: 3527: 3525: 3523: 3520: 3519: 3489: 3477:Main articles: 3475: 3470: 3449: 3446: 3445: 3428: 3424: 3422: 3419: 3418: 3399: 3396: 3395: 3376: 3373: 3372: 3352: 3348: 3346: 3343: 3342: 3320: 3317: 3316: 3294: 3291: 3290: 3273: 3269: 3267: 3264: 3263: 3246: 3242: 3240: 3237: 3236: 3207: 3203: 3198: 3195: 3194: 3178: 3175: 3174: 3154: 3150: 3141: 3137: 3132: 3129: 3128: 3099: 3095: 3087: 3084: 3083: 3058: 3055: 3054: 3029: 3018: 3017: 3016: 3005: 2994: 2993: 2992: 2984: 2977: 2973: 2967: 2956: 2952: 2947: 2944: 2943: 2916: 2905: 2904: 2903: 2894: 2883: 2882: 2881: 2877: 2869: 2867: 2860: 2856: 2836: 2825: 2824: 2823: 2811: 2800: 2799: 2798: 2794: 2786: 2784: 2771: 2767: 2753: 2750: 2749: 2729: 2726: 2725: 2697: 2694: 2693: 2673: 2669: 2658: 2647: 2646: 2645: 2637: 2630: 2626: 2620: 2614: 2611: 2610: 2593: 2589: 2580: 2576: 2567: 2563: 2561: 2558: 2557: 2540: 2536: 2534: 2531: 2530: 2489: 2488: 2480: 2473: 2469: 2467: 2464: 2463: 2447: 2444: 2443: 2413: 2412: 2404: 2397: 2393: 2391: 2388: 2387: 2364: 2360: 2335: 2334: 2330: 2322: 2320: 2319: 2315: 2303: 2299: 2274: 2273: 2269: 2261: 2259: 2258: 2254: 2252: 2249: 2248: 2229: 2226: 2225: 2208: 2204: 2202: 2199: 2198: 2181: 2177: 2175: 2172: 2171: 2138: 2137: 2133: 2125: 2123: 2097: 2096: 2092: 2084: 2082: 2055: 2053: 2049: 2045: 2031: 2030: 2026: 2018: 2016: 1999: 1998: 1981: 1980: 1976: 1968: 1966: 1940: 1937: 1936: 1914: 1913: 1904: 1903: 1892: 1891: 1883: 1871: 1870: 1859: 1856: 1855: 1833: 1813: 1812: 1798: 1797: 1795: 1792: 1791: 1763: 1755: 1752: 1751: 1735: 1732: 1731: 1712: 1709: 1708: 1689: 1681: 1678: 1677: 1655: 1654: 1652: 1649: 1648: 1626: 1625: 1623: 1620: 1619: 1603: 1600: 1599: 1574: 1571: 1570: 1569:and the action 1554: 1551: 1550: 1534: 1531: 1530: 1511: 1508: 1507: 1487: 1483: 1474: 1470: 1465: 1462: 1461: 1445: 1442: 1441: 1422: 1419: 1418: 1414: 1407: 1400: 1384: 1381: 1380: 1356: 1353: 1352: 1348:. That is, the 1320: 1317: 1316: 1276: 1273: 1272: 1256: 1253: 1252: 1241: 1225: 1192:Noether current 1158:and vice versa. 1136: 1101:linear momentum 1076: 990: 961: 960: 946:Integration Bee 921: 918: 911: 910: 886: 883: 876: 875: 848: 845: 838: 837: 819:Volume integral 754: 749: 742: 741: 658: 653: 646: 645: 615: 534: 529: 522: 521: 513:Risch algorithm 488:Euler's formula 374: 369: 362: 361: 343:General Leibniz 226:generalizations 208: 203: 196: 182:Rolle's theorem 177: 152: 88: 82: 77: 71: 68: 67: 35: 28: 23: 22: 15: 12: 11: 5: 23854: 23844: 23843: 23838: 23833: 23828: 23823: 23818: 23813: 23808: 23794: 23793: 23752: 23737: 23731: 23718: 23691:hep-th/0602190 23666: 23660: 23645: 23612: 23583: 23560: 23545: 23493: 23482: 23481:External links 23479: 23478: 23477: 23471: 23451: 23445: 23421: 23400: 23390: 23370: 23346: 23332: 23301: 23298: 23296: 23295: 23214: 23207: 23187: 23180: 23152: 23145: 23119: 23112: 23084: 23077: 23049: 23036: 23029: 23009: 22982:(4): 293–302. 22962: 22947: 22929: 22914: 22894: 22879: 22856: 22832: 22830: 22827: 22825: 22824: 22819: 22814: 22809: 22804: 22802:Gauge symmetry 22799: 22794: 22788: 22787: 22786: 22783:Physics portal 22772: 22756: 22753: 22706: 22705: 22701: 22690: 22679: 22662: 22659: 22651:Ward–Takahashi 22635: 22632: 22627: 22623: 22617: 22613: 22601: 22600: 22585: 22581: 22575: 22571: 22567: 22564: 22561: 22556: 22552: 22548: 22543: 22539: 22533: 22530: 22524: 22518: 22514: 22510: 22506: 22502: 22499: 22496: 22491: 22487: 22481: 22477: 22472: 22468: 22463: 22459: 22455: 22452: 22450: 22448: 22443: 22439: 22435: 22432: 22429: 22426: 22423: 22419: 22413: 22405: 22402: 22397: 22393: 22389: 22386: 22382: 22376: 22372: 22369: 22367: 22363: 22359: 22355: 22354: 22340: 22339: 22328: 22323: 22316: 22312: 22308: 22303: 22299: 22284: 22283: 22271: 22265: 22258: 22254: 22249: 22243: 22239: 22235: 22231: 22225: 22221: 22215: 22211: 22207: 22204: 22201: 22196: 22192: 22188: 22183: 22179: 22173: 22169: 22163: 22160: 22154: 22148: 22144: 22129: 22128: 22117: 22113: 22109: 22106: 22103: 22098: 22094: 22088: 22084: 22079: 22073: 22069: 22065: 22062: 22059: 22055: 22051: 22046: 22042: 22038: 22035: 22030: 22026: 22020: 22016: 22010: 22006: 22002: 21999: 21994: 21990: 21985: 21981: 21976: 21972: 21968: 21965: 21960: 21955: 21952: 21929: 21918: 21917: 21906: 21903: 21900: 21897: 21894: 21891: 21888: 21885: 21882: 21879: 21874: 21870: 21864: 21860: 21856: 21853: 21850: 21847: 21844: 21841: 21838: 21835: 21817: 21816: 21801: 21796: 21792: 21787: 21781: 21777: 21773: 21770: 21767: 21762: 21758: 21754: 21749: 21745: 21739: 21736: 21730: 21726: 21723: 21720: 21718: 21716: 21713: 21708: 21704: 21699: 21695: 21692: 21689: 21686: 21681: 21677: 21673: 21670: 21667: 21664: 21661: 21657: 21651: 21646: 21643: 21640: 21638: 21636: 21633: 21630: 21625: 21620: 21619: 21592: 21589: 21588: 21587: 21576: 21573: 21568: 21565: 21558: 21553: 21550: 21541: 21538: 21532: 21529: 21523: 21520: 21517: 21511: 21508: 21500: 21497: 21491: 21463: 21460: 21453: 21450: 21419: 21416: 21402: 21401: 21384: 21381: 21374: 21371: 21364: 21361: 21358: 21352: 21349: 21343: 21340: 21338: 21336: 21332: 21326: 21319: 21316: 21307: 21303: 21299: 21296: 21291: 21284: 21279: 21276: 21265: 21261: 21256: 21250: 21246: 21242: 21239: 21237: 21235: 21229: 21226: 21220: 21216: 21211: 21204: 21201: 21194: 21187: 21184: 21178: 21174: 21168: 21158: 21151: 21146: 21143: 21134: 21130: 21124: 21118: 21114: 21110: 21107: 21105: 21100: 21097: 21091: 21090: 21076: 21075: 21064: 21059: 21052: 21049: 21040: 21036: 21030: 21026: 21022: 21016: 21013: 20998:so we can set 20985: 20980: 20976: 20970: 20966: 20960: 20956: 20949: 20946: 20942: 20929: 20928: 20913: 20908: 20903: 20896: 20893: 20884: 20880: 20874: 20870: 20866: 20863: 20861: 20859: 20855: 20849: 20842: 20839: 20832: 20827: 20820: 20817: 20809: 20803: 20800: 20796: 20790: 20786: 20782: 20777: 20774: 20771: 20767: 20763: 20758: 20753: 20746: 20743: 20734: 20730: 20724: 20720: 20716: 20713: 20711: 20709: 20704: 20699: 20694: 20690: 20686: 20685: 20671: 20670: 20659: 20654: 20649: 20645: 20641: 20638: 20634: 20630: 20627: 20624: 20619: 20614: 20610: 20605: 20599: 20595: 20568: 20565: 20539: 20528: 20527: 20512: 20509: 20506: 20502: 20497: 20491: 20484: 20481: 20474: 20469: 20462: 20459: 20451: 20445: 20442: 20438: 20432: 20429: 20426: 20422: 20418: 20413: 20408: 20403: 20396: 20391: 20388: 20379: 20372: 20367: 20363: 20355: 20350: 20347: 20344: 20340: 20335: 20331: 20328: 20325: 20323: 20321: 20318: 20315: 20311: 20307: 20304: 20301: 20295: 20290: 20287: 20279: 20276: 20273: 20270: 20264: 20261: 20254: 20248: 20243: 20240: 20237: 20235: 20232: 20226: 20223: 20217: 20211: 20206: 20205: 20190: 20187: 20179: 20178: 20167: 20164: 20158: 20153: 20149: 20137: 20130: 20127: 20120: 20115: 20112: 20104: 20099: 20096: 20093: 20089: 20062: 20059: 20056: 20053: 20049: 20045: 20042: 20031: 20030: 20015: 20012: 20009: 20006: 20003: 20000: 19995: 19990: 19983: 19980: 19971: 19967: 19961: 19958: 19953: 19950: 19948: 19946: 19942: 19938: 19935: 19932: 19929: 19926: 19921: 19916: 19909: 19906: 19897: 19893: 19887: 19884: 19878: 19874: 19869: 19864: 19857: 19854: 19845: 19841: 19837: 19834: 19831: 19829: 19827: 19824: 19821: 19818: 19813: 19809: 19805: 19802: 19794: 19787: 19784: 19777: 19772: 19769: 19761: 19756: 19753: 19750: 19746: 19742: 19739: 19737: 19735: 19732: 19731: 19717: 19716: 19705: 19702: 19699: 19696: 19693: 19690: 19685: 19680: 19673: 19670: 19661: 19657: 19651: 19648: 19643: 19640: 19628:so we can set 19626: 19625: 19612: 19605: 19602: 19590: 19586: 19582: 19577: 19574: 19571: 19568: 19565: 19557: 19553: 19549: 19544: 19537: 19534: 19525: 19518: 19515: 19506: 19502: 19498: 19495: 19491: 19487: 19484: 19481: 19478: 19475: 19470: 19465: 19458: 19455: 19446: 19442: 19436: 19433: 19427: 19420: 19417: 19413: 19408: 19405: 19402: 19399: 19396: 19365: 19362: 19359: 19353: 19350: 19344: 19341: 19338: 19335: 19332: 19329: 19326: 19323: 19293:kinetic energy 19289: 19288: 19273: 19270: 19267: 19262: 19258: 19255: 19252: 19249: 19246: 19243: 19240: 19237: 19232: 19227: 19220: 19217: 19208: 19203: 19200: 19197: 19193: 19187: 19184: 19178: 19174: 19171: 19168: 19166: 19164: 19161: 19158: 19153: 19149: 19146: 19143: 19137: 19134: 19128: 19125: 19122: 19119: 19116: 19112: 19108: 19105: 19102: 19099: 19097: 19095: 19092: 19089: 19084: 19079: 19078: 19039: 19036: 19034: 19031: 19030: 19029: 19018: 19015: 19011: 19007: 19003: 18999: 18995: 18992: 18989: 18986: 18982: 18976: 18968: 18965: 18960: 18956: 18950: 18946: 18942: 18939: 18935: 18929: 18924: 18918: 18914: 18910: 18907: 18904: 18901: 18898: 18893: 18889: 18884: 18878: 18870: 18867: 18862: 18858: 18852: 18848: 18844: 18841: 18837: 18831: 18827: 18824: 18821: 18818: 18815: 18812: 18808: 18802: 18794: 18791: 18786: 18782: 18778: 18775: 18771: 18765: 18761: 18756: 18752: 18747: 18741: 18737: 18710: 18699: 18698: 18683: 18678: 18674: 18668: 18664: 18661: 18658: 18655: 18651: 18645: 18637: 18634: 18629: 18625: 18621: 18618: 18614: 18608: 18604: 18599: 18595: 18590: 18584: 18580: 18576: 18573: 18570: 18567: 18565: 18563: 18560: 18555: 18551: 18545: 18541: 18538: 18535: 18532: 18528: 18522: 18513: 18509: 18504: 18500: 18495: 18491: 18487: 18481: 18477: 18472: 18468: 18464: 18461: 18456: 18451: 18448: 18445: 18441: 18437: 18434: 18431: 18429: 18427: 18424: 18419: 18415: 18410: 18406: 18403: 18400: 18397: 18394: 18391: 18388: 18383: 18379: 18374: 18368: 18360: 18357: 18352: 18348: 18344: 18341: 18337: 18331: 18327: 18324: 18321: 18318: 18315: 18312: 18308: 18302: 18294: 18291: 18287: 18281: 18276: 18272: 18269: 18266: 18264: 18262: 18259: 18254: 18250: 18246: 18241: 18236: 18233: 18230: 18227: 18224: 18221: 18218: 18215: 18213: 18211: 18206: 18201: 18198: 18195: 18192: 18189: 18186: 18185: 18171: 18170: 18157: 18153: 18147: 18143: 18139: 18136: 18131: 18126: 18123: 18100: 18005: 18002: 17987: 17982: 17977: 17973: 17969: 17966: 17963: 17960: 17957: 17954: 17949: 17945: 17941: 17936: 17932: 17928: 17925: 17922: 17919: 17916: 17911: 17907: 17903: 17898: 17894: 17890: 17886: 17880: 17872: 17869: 17864: 17860: 17856: 17853: 17849: 17843: 17839: 17834: 17829: 17825: 17812: 17805: 17798: 17781: 17776: 17772: 17766: 17762: 17758: 17755: 17752: 17747: 17742: 17737: 17733: 17729: 17724: 17720: 17716: 17713: 17710: 17705: 17700: 17695: 17691: 17687: 17682: 17678: 17674: 17671: 17666: 17661: 17658: 17653: 17649: 17645: 17640: 17636: 17632: 17608: 17603: 17599: 17593: 17589: 17585: 17582: 17577: 17572: 17567: 17563: 17540: 17535: 17531: 17525: 17521: 17517: 17514: 17509: 17504: 17499: 17495: 17482: 17475: 17468: 17465: 17444: 17441: 17437: 17426: 17423: 17417: 17412: 17408: 17404: 17400: 17388: 17387: 17376: 17373: 17368: 17364: 17360: 17355: 17351: 17345: 17342: 17338: 17317: 17314: 17286: 17285: 17274: 17269: 17265: 17261: 17258: 17255: 17252: 17249: 17243: 17240: 17235: 17231: 17227: 17224: 17217: 17212: 17205: 17199: 17195: 17169: 17165: 17149: 17148: 17137: 17134: 17130: 17124: 17120: 17116: 17113: 17110: 17107: 17104: 17098: 17095: 17090: 17086: 17082: 17079: 17072: 17067: 17060: 17054: 17050: 17031: 17030: 17015: 17010: 17006: 17001: 16994: 16990: 16984: 16981: 16977: 16973: 16970: 16968: 16966: 16961: 16957: 16952: 16947: 16944: 16941: 16938: 16932: 16929: 16924: 16920: 16916: 16913: 16906: 16901: 16893: 16890: 16886: 16882: 16879: 16874: 16869: 16863: 16860: 16857: 16854: 16850: 16843: 16840: 16835: 16831: 16827: 16824: 16817: 16812: 16804: 16800: 16796: 16790: 16787: 16780: 16775: 16768: 16762: 16758: 16754: 16751: 16749: 16746: 16742: 16737: 16732: 16724: 16717: 16713: 16708: 16704: 16701: 16700: 16682:Euler–Lagrange 16639:If this holds 16637: 16636: 16625: 16622: 16619: 16616: 16613: 16610: 16607: 16604: 16599: 16595: 16591: 16588: 16585: 16582: 16579: 16576: 16571: 16566: 16563: 16560: 16557: 16552: 16532: 16531: 16520: 16517: 16514: 16509: 16505: 16499: 16495: 16491: 16488: 16485: 16482: 16479: 16474: 16469: 16466: 16448: 16447: 16434: 16430: 16426: 16422: 16419: 16416: 16413: 16410: 16407: 16404: 16401: 16398: 16395: 16392: 16389: 16386: 16383: 16380: 16375: 16371: 16365: 16362: 16358: 16354: 16350: 16346: 16341: 16336: 16328: 16321: 16317: 16312: 16308: 16272: 16235: 16224: 16223: 16212: 16209: 16203: 16200: 16197: 16194: 16191: 16186: 16183: 16180: 16175: 16170: 16144: 16122: 16107:such that all 16096: 16074: 16044: 16012: 15997: 15996: 15985: 15982: 15977: 15973: 15968: 15965: 15962: 15959: 15956: 15953: 15950: 15945: 15941: 15937: 15934: 15931: 15928: 15925: 15922: 15917: 15910: 15906: 15901: 15897: 15894: 15891: 15886: 15860: 15838: 15814: 15799: 15798: 15787: 15784: 15781: 15778: 15773: 15769: 15765: 15762: 15759: 15754: 15740:of a function 15721: 15718: 15715: 15710: 15680: 15670:, rather than 15658: 15642: 15641: 15630: 15626: 15622: 15617: 15612: 15607: 15588: 15587: 15573: 15568: 15540: 15535: 15511: 15507: 15503: 15500: 15497: 15492: 15488: 15453: 15432: 15362: 15332: 15329: 15328: 15327: 15316: 15310: 15306: 15301: 15293: 15290: 15283: 15279: 15273: 15268: 15265: 15259: 15255: 15251: 15245: 15241: 15236: 15233: 15228: 15224: 15218: 15212: 15201: 15198: 15191: 15187: 15181: 15176: 15173: 15166: 15162: 15157: 15153: 15138: 15137: 15126: 15123: 15118: 15114: 15105: 15101: 15097: 15093: 15069: 15068: 15057: 15051: 15047: 15041: 15035: 15029: 15026: 15021: 15017: 15013: 15010: 15005: 15001: 14994: 14991: 14974: 14973: 14962: 14956: 14952: 14943: 14939: 14935: 14928: 14924: 14920: 14914: 14909: 14905: 14899: 14893: 14867: 14864: 14859: 14856: 14849: 14845: 14820: 14816: 14789: 14785: 14773:Lie derivative 14758: 14754: 14748: 14742: 14728: 14727: 14710: 14706: 14700: 14694: 14688: 14685: 14680: 14676: 14669: 14666: 14660: 14655: 14651: 14647: 14644: 14641: 14639: 14635: 14631: 14627: 14624: 14623: 14618: 14614: 14610: 14607: 14604: 14602: 14598: 14594: 14590: 14587: 14586: 14568: 14567: 14556: 14552: 14549: 14545: 14539: 14535: 14531: 14527: 14521: 14517: 14513: 14508: 14505: 14498: 14494: 14488: 14483: 14479: 14474: 14470: 14467: 14462: 14458: 14451: 14448: 14437: 14434: 14427: 14423: 14417: 14412: 14409: 14402: 14393: 14389: 14385: 14381: 14365: 14364: 14353: 14349: 14346: 14343: 14338: 14334: 14329: 14323: 14319: 14315: 14311: 14305: 14301: 14297: 14292: 14289: 14282: 14278: 14272: 14267: 14263: 14258: 14254: 14251: 14246: 14242: 14235: 14232: 14221: 14218: 14211: 14207: 14201: 14196: 14193: 14186: 14177: 14173: 14169: 14165: 14158: 14154: 14139: 14138: 14123: 14119: 14113: 14109: 14102: 14099: 14088: 14085: 14078: 14074: 14068: 14063: 14060: 14053: 14044: 14040: 14036: 14032: 14027: 14022: 14019: 14012: 14008: 13999: 13996: 13985: 13982: 13975: 13971: 13965: 13960: 13957: 13951: 13946: 13942: 13935: 13932: 13925: 13917: 13914: 13907: 13903: 13897: 13892: 13889: 13883: 13874: 13870: 13866: 13862: 13857: 13853: 13850: 13849: 13845: 13840: 13834: 13830: 13826: 13821: 13818: 13811: 13807: 13801: 13796: 13792: 13787: 13783: 13780: 13776: 13770: 13766: 13762: 13757: 13754: 13747: 13743: 13737: 13732: 13728: 13723: 13719: 13715: 13711: 13709: 13695: 13694: 13678: 13674: 13670: 13665: 13662: 13656: 13652: 13644: 13641: 13634: 13630: 13624: 13619: 13616: 13610: 13601: 13597: 13593: 13589: 13573: 13572: 13561: 13556: 13550: 13546: 13539: 13536: 13529: 13520: 13516: 13512: 13508: 13503: 13495: 13491: 13487: 13480: 13476: 13472: 13463: 13460: 13454: 13449: 13446: 13439: 13435: 13426: 13423: 13403: 13402: 13391: 13385: 13382: 13375: 13371: 13362: 13359: 13348: 13345: 13338: 13334: 13328: 13323: 13320: 13314: 13309: 13305: 13298: 13295: 13284: 13280: 13276: 13271: 13268: 13262: 13258: 13253: 13247: 13243: 13239: 13234: 13231: 13224: 13220: 13214: 13209: 13205: 13200: 13196: 13193: 13189: 13183: 13179: 13175: 13170: 13167: 13160: 13156: 13150: 13145: 13141: 13136: 13132: 13128: 13113: 13112: 13101: 13097: 13094: 13091: 13086: 13082: 13077: 13072: 13066: 13062: 13058: 13054: 13048: 13044: 13040: 13035: 13032: 13025: 13021: 13015: 13010: 13006: 13001: 12997: 12993: 12984: 12980: 12976: 12972: 12967: 12963: 12958: 12952: 12948: 12944: 12939: 12936: 12929: 12925: 12919: 12914: 12910: 12905: 12901: 12898: 12894: 12888: 12884: 12880: 12875: 12872: 12865: 12861: 12855: 12850: 12846: 12841: 12837: 12833: 12828: 12822: 12818: 12799: 12798: 12787: 12778: 12774: 12770: 12763: 12759: 12755: 12749: 12744: 12741: 12734: 12730: 12713: 12712: 12701: 12698: 12695: 12690: 12686: 12681: 12675: 12671: 12667: 12662: 12659: 12652: 12648: 12642: 12637: 12633: 12628: 12624: 12619: 12615: 12611: 12608: 12603: 12599: 12594: 12588: 12584: 12580: 12575: 12572: 12565: 12561: 12555: 12550: 12546: 12541: 12537: 12530: 12526: 12521: 12498: 12497: 12486: 12481: 12476: 12472: 12468: 12462: 12458: 12451: 12448: 12442: 12438: 12433: 12429: 12425: 12419: 12415: 12411: 12407: 12402: 12398: 12394: 12388: 12384: 12350: 12346: 12342: 12331: 12330: 12319: 12314: 12309: 12305: 12301: 12295: 12291: 12287: 12284: 12280: 12275: 12271: 12267: 12261: 12257: 12253: 12249: 12244: 12240: 12236: 12230: 12226: 12222: 12217: 12213: 12198: 12197: 12184: 12180: 12176: 12173: 12168: 12164: 12160: 12155: 12151: 12147: 12142: 12138: 12092: 12088: 12075: 12072: 12068: 12067: 12056: 12050: 12047: 12042: 12039: 12027: 12023: 12016: 12011: 12008: 12002: 11999: 11995: 11991: 11988: 11982: 11978: 11965: 11961: 11954: 11949: 11946: 11939: 11915: 11912: 11905: 11901: 11896: 11893: 11875: 11874: 11860: 11857: 11852: 11849: 11837: 11833: 11826: 11821: 11818: 11812: 11809: 11805: 11801: 11798: 11792: 11788: 11777: 11773: 11768: 11765: 11753: 11749: 11742: 11737: 11734: 11727: 11712: 11711: 11696: 11693: 11688: 11684: 11680: 11674: 11671: 11666: 11663: 11651: 11647: 11640: 11635: 11632: 11626: 11623: 11618: 11614: 11610: 11604: 11601: 11596: 11593: 11581: 11577: 11570: 11565: 11562: 11556: 11551: 11549: 11547: 11544: 11541: 11536: 11532: 11528: 11522: 11518: 11507: 11503: 11498: 11495: 11483: 11479: 11472: 11467: 11464: 11458: 11455: 11452: 11447: 11443: 11439: 11433: 11429: 11418: 11414: 11409: 11406: 11394: 11390: 11383: 11378: 11375: 11369: 11366: 11363: 11358: 11354: 11350: 11347: 11342: 11338: 11334: 11328: 11324: 11317: 11314: 11309: 11305: 11301: 11297: 11293: 11290: 11287: 11284: 11281: 11276: 11272: 11268: 11265: 11260: 11256: 11252: 11246: 11242: 11235: 11232: 11227: 11223: 11219: 11215: 11211: 11208: 11205: 11201: 11198: 11195: 11194: 11180: 11179: 11168: 11162: 11158: 11147: 11143: 11140: 11137: 11132: 11127: 11123: 11110: 11106: 11099: 11094: 11091: 11085: 11079: 11076: 11071: 11068: 11058: 11054: 11049: 11046: 11040: 11034: 11030: 11019: 11015: 11012: 11009: 11004: 10999: 10995: 10982: 10978: 10971: 10966: 10963: 10957: 10951: 10948: 10943: 10940: 10933: 10923: 10919: 10912: 10907: 10904: 10895: 10892: 10888: 10882: 10878: 10874: 10867: 10864: 10859: 10856: 10844: 10840: 10833: 10828: 10825: 10818: 10811: 10808: 10804: 10788: 10787: 10772: 10769: 10766: 10759: 10755: 10744: 10740: 10737: 10734: 10729: 10724: 10720: 10707: 10703: 10696: 10691: 10688: 10682: 10676: 10673: 10668: 10665: 10655: 10651: 10646: 10643: 10633: 10629: 10621: 10617: 10612: 10608: 10603: 10601: 10599: 10596: 10593: 10588: 10584: 10580: 10574: 10570: 10559: 10555: 10550: 10547: 10535: 10531: 10524: 10519: 10516: 10510: 10507: 10504: 10499: 10495: 10491: 10485: 10481: 10470: 10466: 10461: 10458: 10446: 10442: 10435: 10430: 10427: 10421: 10418: 10415: 10410: 10406: 10402: 10399: 10394: 10390: 10386: 10380: 10376: 10369: 10366: 10361: 10357: 10353: 10349: 10345: 10342: 10339: 10336: 10333: 10328: 10324: 10320: 10317: 10312: 10308: 10304: 10298: 10294: 10287: 10284: 10279: 10275: 10271: 10267: 10263: 10260: 10257: 10253: 10250: 10247: 10244: 10238: 10235: 10229: 10226: 10222: 10216: 10213: 10210: 10209: 10195: 10194: 10179: 10176: 10169: 10165: 10154: 10150: 10145: 10142: 10130: 10126: 10119: 10114: 10111: 10105: 10102: 10096: 10092: 10084: 10077: 10073: 10061: 10057: 10052: 10048: 10045: 10040: 10035: 10031: 10023: 10013: 10009: 10002: 9997: 9994: 9988: 9985: 9979: 9975: 9964: 9960: 9955: 9952: 9942: 9938: 9933: 9930: 9924: 9921: 9919: 9917: 9914: 9907: 9903: 9892: 9888: 9883: 9880: 9868: 9864: 9857: 9852: 9849: 9843: 9840: 9834: 9830: 9822: 9814: 9810: 9805: 9802: 9793: 9790: 9786: 9780: 9770: 9766: 9759: 9754: 9751: 9745: 9742: 9736: 9732: 9721: 9717: 9712: 9709: 9702: 9692: 9688: 9681: 9676: 9673: 9664: 9661: 9657: 9651: 9647: 9644: 9642: 9639: 9635: 9629: 9625: 9614: 9610: 9605: 9602: 9590: 9586: 9579: 9574: 9571: 9564: 9557: 9554: 9550: 9545: 9544: 9530: 9529: 9514: 9511: 9508: 9503: 9499: 9493: 9489: 9478: 9474: 9469: 9466: 9460: 9454: 9450: 9439: 9435: 9432: 9429: 9424: 9419: 9415: 9408: 9405: 9400: 9393: 9389: 9376: 9372: 9367: 9363: 9360: 9355: 9350: 9346: 9339: 9335: 9325: 9321: 9314: 9309: 9306: 9300: 9296: 9289: 9286: 9281: 9278: 9272: 9269: 9263: 9259: 9248: 9244: 9239: 9236: 9230: 9226: 9218: 9214: 9209: 9206: 9196: 9192: 9184: 9180: 9175: 9171: 9166: 9164: 9162: 9159: 9156: 9151: 9147: 9143: 9140: 9135: 9131: 9127: 9121: 9117: 9110: 9107: 9102: 9098: 9094: 9090: 9086: 9083: 9080: 9077: 9074: 9069: 9065: 9061: 9058: 9053: 9049: 9045: 9039: 9035: 9028: 9025: 9020: 9016: 9012: 9008: 9004: 9001: 8998: 8994: 8991: 8988: 8985: 8979: 8976: 8970: 8967: 8963: 8957: 8954: 8951: 8950: 8936:Leibniz's rule 8934:= 0 and using 8924: 8923: 8907: 8904: 8900: 8896: 8892: 8889: 8885: 8882: 8879: 8876: 8873: 8869: 8866: 8862: 8856: 8852: 8845: 8842: 8839: 8836: 8833: 8830: 8827: 8823: 8820: 8816: 8812: 8808: 8801: 8797: 8792: 8789: 8783: 8780: 8777: 8774: 8771: 8768: 8765: 8762: 8758: 8755: 8751: 8747: 8743: 8740: 8737: 8734: 8729: 8726: 8723: 8718: 8714: 8708: 8705: 8702: 8697: 8693: 8688: 8684: 8681: 8679: 8677: 8673: 8670: 8666: 8662: 8658: 8655: 8651: 8648: 8644: 8641: 8637: 8633: 8627: 8623: 8615: 8612: 8608: 8605: 8601: 8597: 8593: 8588: 8585: 8580: 8577: 8574: 8569: 8565: 8559: 8556: 8553: 8548: 8544: 8539: 8535: 8532: 8530: 8528: 8525: 8522: 8518: 8515: 8511: 8510: 8496: 8495: 8484: 8481: 8478: 8475: 8472: 8468: 8465: 8461: 8455: 8451: 8444: 8441: 8438: 8435: 8432: 8429: 8426: 8422: 8419: 8415: 8411: 8407: 8400: 8396: 8391: 8388: 8382: 8379: 8376: 8373: 8370: 8367: 8364: 8360: 8356: 8353: 8347: 8344: 8340: 8335: 8332: 8328: 8325: 8321: 8317: 8311: 8307: 8299: 8296: 8293: 8290: 8284: 8280: 8254: 8253: 8238: 8235: 8232: 8229: 8226: 8223: 8220: 8216: 8213: 8209: 8205: 8201: 8198: 8195: 8192: 8189: 8186: 8183: 8180: 8177: 8173: 8169: 8166: 8163: 8160: 8156: 8153: 8149: 8145: 8141: 8136: 8133: 8131: 8129: 8126: 8123: 8119: 8115: 8114: 8111: 8108: 8105: 8102: 8099: 8095: 8092: 8088: 8085: 8083: 8081: 8078: 8077: 8055: 8054: 8043: 8040: 8037: 8034: 8027: 8023: 8018: 8015: 8009: 8006: 8003: 8000: 7991: 7987: 7980: 7975: 7972: 7963: 7960: 7956: 7926: 7923: 7919: 7916: 7913: 7910: 7907: 7904: 7898: 7894: 7887: 7884: 7881: 7878: 7874: 7870: 7867: 7860: 7856: 7848: 7844: 7839: 7835: 7832: 7814: 7811: 7809: 7806: 7794: 7793: 7782: 7774: 7771: 7767: 7762: 7758: 7750: 7746: 7742: 7735: 7731: 7727: 7721: 7716: 7712: 7703: 7699: 7695: 7690: 7687: 7680: 7676: 7673: 7668: 7664: 7645: 7644: 7633: 7628: 7624: 7620: 7615: 7611: 7607: 7602: 7599: 7595: 7589: 7585: 7579: 7575: 7571: 7566: 7562: 7558: 7555: 7501: 7500: 7489: 7481: 7477: 7471: 7468: 7465: 7461: 7457: 7452: 7448: 7441: 7435: 7430: 7427: 7423: 7419: 7416: 7392:field and its 7378:complex number 7323: 7303: 7283: 7272: 7271: 7260: 7257: 7252: 7249: 7240: 7231: 7227: 7212: 7211: 7198: 7191: 7186: 7182: 7178: 7173: 7170: 7166: 7162: 7158: 7150: 7147: 7143: 7139: 7132: 7127: 7121: 7117: 7109: 7106: 7102: 7098: 7091: 7086: 7080: 7075: 7071: 7065: 7060: 7056: 7052: 7047: 7040: 7035: 7031: 7027: 7024: 7019: 7010: 7006: 6982: 6962: 6942: 6922: 6908: 6899: 6898: 6885: 6878: 6873: 6869: 6865: 6862: 6857: 6852: 6848: 6823: 6817: 6812: 6808: 6802: 6798: 6794: 6789: 6785: 6780: 6774: 6769: 6765: 6760: 6756: 6752: 6746: 6733: 6732: 6721: 6718: 6713: 6709: 6703: 6698: 6694: 6690: 6687: 6682: 6678: 6652: 6648: 6625: 6621: 6617: 6614: 6609: 6605: 6582: 6578: 6556: 6550: 6545: 6541: 6535: 6531: 6527: 6522: 6518: 6513: 6509: 6506: 6503: 6498: 6494: 6490: 6487: 6463: 6441: 6436: 6432: 6426: 6422: 6418: 6413: 6409: 6405: 6400: 6396: 6370: 6364: 6360: 6356: 6352: 6346: 6342: 6338: 6334: 6329: 6325: 6307: 6306: 6295: 6292: 6289: 6284: 6280: 6274: 6270: 6239: 6219: 6200: 6199: 6186: 6182: 6178: 6170: 6167: 6163: 6159: 6152: 6147: 6141: 6136: 6131: 6127: 6123: 6118: 6113: 6109: 6082: 6071: 6070: 6059: 6054: 6049: 6045: 6039: 6035: 6029: 6025: 6021: 6016: 6011: 6006: 5990: 5989: 5978: 5973: 5969: 5963: 5959: 5955: 5952: 5949: 5946: 5923: 5920: 5917: 5914: 5911: 5908: 5905: 5902: 5899: 5896: 5881: 5880: 5869: 5864: 5860: 5854: 5850: 5846: 5843: 5838: 5833: 5828: 5801: 5777: 5755: 5735: 5713: 5709: 5688: 5677: 5676: 5665: 5662: 5659: 5656: 5653: 5650: 5647: 5624: 5602: 5601: 5590: 5585: 5581: 5575: 5569: 5565: 5561: 5558: 5553: 5549: 5545: 5542: 5538: 5532: 5527: 5524: 5519: 5490: 5487: 5484: 5480: 5476: 5473: 5453: 5426: 5423: 5421:is conserved. 5399: 5398: 5387: 5383: 5379: 5375: 5371: 5367: 5362: 5358: 5354: 5349: 5345: 5341: 5337: 5333: 5328: 5324: 5320: 5315: 5311: 5307: 5303: 5299: 5295: 5286: 5282: 5275: 5270: 5267: 5250: 5249: 5238: 5234: 5230: 5226: 5222: 5218: 5175: 5174: 5163: 5159: 5155: 5151: 5146: 5143: 5140: 5136: 5132: 5128: 5107:about an axis 5052: 5051: 5040: 5031: 5026: 5022: 5015: 5010: 5007: 5001: 4996: 4992: 4975: 4963: 4946: 4937: 4928: 4919: 4908: 4907: 4896: 4893: 4890: 4884: 4880: 4873: 4864: 4860: 4853: 4848: 4845: 4839: 4836: 4782: 4779: 4771: 4770: 4757: 4752: 4747: 4738: 4734: 4727: 4722: 4719: 4713: 4708: 4704: 4699: 4695: 4692: 4686: 4682: 4675: 4666: 4662: 4655: 4650: 4647: 4640: 4599: 4593: 4592: 4586: 4578: 4575:time evolution 4569: 4551: 4545: 4544: 4529: 4521: 4516: 4509: 4505: 4499: 4495: 4491: 4488: 4486: 4483: 4479: 4476: 4475: 4470: 4466: 4460: 4456: 4450: 4446: 4442: 4439: 4437: 4435: 4432: 4429: 4428: 4384: 4383: 4368: 4361: 4357: 4354: 4350: 4346: 4341: 4336: 4331: 4328: 4326: 4323: 4319: 4318: 4315: 4312: 4309: 4306: 4303: 4298: 4294: 4290: 4287: 4285: 4283: 4280: 4279: 4246: 4243: 4235: 4232: 4195: 4188: 4176: 4175: 4159: 4152: 4149: 4142: 4137: 4134: 4128: 4123: 4119: 4104: 4103: 4092: 4086: 4083: 4077: 4074: 4067: 4063: 4059: 4053: 4049: 4041: 4034: 4031: 4024: 4019: 4016: 4010: 4003: 4000: 3996: 3978: 3973: 3972: 3961: 3951: 3947: 3942: 3939: 3933: 3929: 3920: 3916: 3909: 3904: 3901: 3895: 3888: 3885: 3881: 3846: 3845: 3834: 3825: 3822: 3816: 3812: 3806: 3800: 3796: 3770: 3769: 3758: 3752: 3749: 3745: 3742: 3739: 3733: 3729: 3722: 3718: 3714: 3711: 3708: 3705: 3702: 3607:René Descartes 3603:kinetic energy 3583: 3582: 3571: 3565: 3562: 3556: 3553: 3547: 3541: 3538: 3533: 3530: 3474: 3471: 3469: 3468: 3456: 3453: 3431: 3427: 3406: 3403: 3383: 3380: 3370: 3369: 3355: 3351: 3327: 3324: 3313: 3312:of the region, 3301: 3298: 3276: 3272: 3249: 3245: 3235:the endpoints 3233: 3221: 3218: 3215: 3210: 3206: 3202: 3182: 3162: 3157: 3153: 3149: 3144: 3140: 3136: 3113: 3110: 3107: 3102: 3098: 3094: 3091: 3082:entire fields 3071: 3068: 3065: 3062: 3051: 3038: 3032: 3025: 3022: 3014: 3008: 3001: 2998: 2991: 2987: 2983: 2980: 2976: 2970: 2966: 2962: 2959: 2955: 2951: 2929: 2925: 2919: 2912: 2909: 2897: 2890: 2887: 2880: 2875: 2872: 2866: 2863: 2859: 2855: 2852: 2849: 2845: 2839: 2832: 2829: 2822: 2814: 2807: 2804: 2797: 2792: 2789: 2783: 2780: 2777: 2774: 2770: 2766: 2763: 2760: 2757: 2736: 2733: 2713: 2710: 2707: 2704: 2701: 2690: 2676: 2672: 2667: 2661: 2654: 2651: 2644: 2640: 2636: 2633: 2629: 2623: 2619: 2596: 2592: 2588: 2583: 2579: 2575: 2570: 2566: 2543: 2539: 2526: 2510: 2507: 2503: 2496: 2493: 2487: 2483: 2479: 2476: 2472: 2451: 2431: 2427: 2420: 2417: 2411: 2407: 2403: 2400: 2396: 2375: 2372: 2367: 2363: 2359: 2355: 2351: 2342: 2339: 2333: 2328: 2325: 2318: 2314: 2311: 2306: 2302: 2298: 2294: 2290: 2281: 2278: 2272: 2267: 2264: 2257: 2236: 2233: 2211: 2207: 2184: 2180: 2157: 2154: 2145: 2142: 2136: 2131: 2128: 2122: 2119: 2116: 2113: 2104: 2101: 2095: 2090: 2087: 2081: 2075: 2071: 2065: 2061: 2058: 2052: 2048: 2038: 2035: 2029: 2024: 2021: 2015: 2012: 2006: 2003: 1997: 1988: 1985: 1979: 1974: 1971: 1965: 1962: 1959: 1956: 1953: 1950: 1947: 1944: 1921: 1918: 1912: 1907: 1899: 1896: 1890: 1886: 1882: 1879: 1874: 1869: 1866: 1863: 1840: 1836: 1832: 1829: 1826: 1820: 1817: 1811: 1805: 1802: 1779: 1776: 1773: 1770: 1766: 1762: 1759: 1739: 1719: 1716: 1696: 1692: 1688: 1685: 1662: 1659: 1633: 1630: 1607: 1587: 1584: 1581: 1578: 1558: 1538: 1518: 1515: 1495: 1490: 1486: 1482: 1477: 1473: 1469: 1449: 1426: 1412: 1405: 1388: 1360: 1346:laws of motion 1333: 1330: 1327: 1324: 1301: 1298: 1295: 1292: 1289: 1286: 1283: 1280: 1260: 1240: 1237: 1221: 1188:Noether charge 1135: 1132: 1075: 1072: 992: 991: 989: 988: 981: 974: 966: 963: 962: 959: 958: 953: 948: 943: 941:List of topics 938: 933: 928: 922: 917: 916: 913: 912: 909: 908: 903: 898: 893: 887: 882: 881: 878: 877: 872: 871: 870: 869: 864: 859: 849: 844: 843: 840: 839: 834: 833: 832: 831: 826: 821: 816: 811: 806: 801: 793: 792: 788: 787: 786: 785: 780: 775: 770: 762: 761: 755: 748: 747: 744: 743: 738: 737: 736: 735: 730: 725: 720: 715: 710: 702: 701: 697: 696: 695: 694: 689: 684: 679: 674: 669: 659: 652: 651: 648: 647: 642: 641: 640: 639: 634: 629: 624: 619: 613: 608: 603: 598: 593: 585: 584: 578: 577: 576: 575: 570: 565: 560: 555: 550: 535: 528: 527: 524: 523: 518: 517: 516: 515: 510: 505: 500: 498:Changing order 495: 490: 485: 467: 462: 457: 449: 448: 447:Integration by 444: 443: 442: 441: 436: 431: 426: 421: 411: 409:Antiderivative 403: 402: 398: 397: 396: 395: 390: 385: 375: 368: 367: 364: 363: 358: 357: 356: 355: 350: 345: 340: 335: 330: 325: 320: 315: 310: 302: 301: 295: 294: 293: 292: 287: 282: 277: 272: 267: 259: 258: 254: 253: 252: 251: 250: 249: 244: 239: 229: 216: 215: 209: 202: 201: 198: 197: 195: 194: 189: 184: 178: 176: 175: 170: 164: 163: 162: 154: 153: 141: 138: 135: 132: 129: 126: 123: 120: 117: 114: 111: 108: 104: 101: 98: 94: 91: 85: 80: 76: 66: 63: 62: 56: 55: 43:First page of 26: 18:Noether charge 9: 6: 4: 3: 2: 23853: 23842: 23839: 23837: 23834: 23832: 23829: 23827: 23824: 23822: 23819: 23817: 23814: 23812: 23809: 23807: 23804: 23803: 23801: 23790: 23786: 23782: 23778: 23773: 23768: 23764: 23760: 23759: 23753: 23748: 23743: 23738: 23734: 23728: 23724: 23719: 23709:on 2016-03-04 23705: 23701: 23697: 23692: 23687: 23683: 23679: 23672: 23667: 23665:at MathPages. 23664: 23661: 23656: 23651: 23646: 23642: 23638: 23634: 23630: 23627:(4): 428–35. 23626: 23622: 23618: 23613: 23609: 23605: 23601: 23597: 23594:(2): 025025. 23593: 23589: 23584: 23573: 23569: 23565: 23561: 23556: 23551: 23546: 23544:1918:235–257) 23543: 23540:(Original in 23537: 23533: 23529: 23525: 23521: 23517: 23512: 23507: 23503: 23499: 23494: 23490: 23485: 23484: 23474: 23468: 23464: 23460: 23456: 23452: 23448: 23446:0-387-95000-1 23442: 23438: 23434: 23430: 23426: 23422: 23410: 23406: 23401: 23398: 23393: 23387: 23383: 23379: 23375: 23371: 23360: 23359:Union College 23356: 23355:Honors Theses 23352: 23347: 23343: 23339: 23335: 23329: 23325: 23321: 23317: 23313: 23309: 23304: 23303: 23291: 23285: 23277: 23273: 23269: 23265: 23261: 23257: 23253: 23249: 23244: 23243:gr-qc/9503052 23239: 23235: 23231: 23230: 23225: 23222:Iyer, Vivek; 23218: 23210: 23208:0-201-50397-2 23204: 23200: 23199: 23191: 23183: 23181:0-201-02918-9 23177: 23173: 23172: 23167: 23161: 23159: 23157: 23148: 23146:0-486-65067-7 23142: 23138: 23134: 23128: 23126: 23124: 23115: 23109: 23105: 23101: 23097: 23096: 23088: 23080: 23074: 23070: 23066: 23062: 23061: 23053: 23046: 23040: 23032: 23030:0-471-55264-X 23026: 23022: 23021: 23013: 23005: 23001: 22997: 22993: 22989: 22985: 22981: 22977: 22973: 22966: 22958: 22954: 22950: 22944: 22940: 22933: 22925: 22921: 22917: 22915:0-521-57327-0 22911: 22907: 22906: 22898: 22890: 22886: 22882: 22876: 22872: 22871: 22863: 22861: 22852: 22848: 22844: 22837: 22833: 22823: 22820: 22818: 22815: 22813: 22810: 22808: 22805: 22803: 22800: 22798: 22795: 22793: 22790: 22789: 22784: 22778: 22773: 22770: 22764: 22759: 22752: 22750: 22746: 22741: 22739: 22735: 22731: 22727: 22723: 22719: 22715: 22711: 22702: 22699: 22695: 22691: 22688: 22684: 22680: 22677: 22673: 22669: 22668: 22667: 22658: 22656: 22652: 22647: 22633: 22630: 22625: 22621: 22615: 22583: 22579: 22573: 22569: 22565: 22562: 22559: 22554: 22546: 22541: 22531: 22528: 22522: 22516: 22512: 22508: 22504: 22500: 22497: 22494: 22489: 22479: 22475: 22470: 22466: 22461: 22453: 22451: 22441: 22437: 22433: 22427: 22421: 22417: 22400: 22395: 22374: 22370: 22368: 22361: 22357: 22345: 22344: 22343: 22326: 22314: 22310: 22306: 22301: 22297: 22289: 22288: 22287: 22269: 22256: 22252: 22247: 22241: 22233: 22229: 22223: 22219: 22213: 22209: 22205: 22202: 22199: 22194: 22186: 22181: 22171: 22167: 22161: 22158: 22152: 22146: 22134: 22133: 22132: 22115: 22111: 22107: 22104: 22101: 22096: 22086: 22082: 22077: 22071: 22067: 22063: 22060: 22057: 22053: 22049: 22044: 22036: 22033: 22028: 22018: 22008: 22004: 22000: 21997: 21992: 21983: 21979: 21974: 21966: 21950: 21943: 21942: 21941: 21927: 21904: 21898: 21892: 21889: 21883: 21877: 21872: 21862: 21858: 21854: 21845: 21839: 21833: 21826: 21825: 21824: 21822: 21799: 21794: 21790: 21785: 21779: 21775: 21771: 21768: 21765: 21760: 21752: 21747: 21737: 21734: 21728: 21724: 21721: 21719: 21711: 21706: 21702: 21697: 21690: 21684: 21679: 21671: 21665: 21659: 21655: 21644: 21641: 21639: 21631: 21610: 21609: 21608: 21606: 21602: 21598: 21574: 21571: 21566: 21563: 21556: 21548: 21539: 21536: 21527: 21518: 21515: 21509: 21506: 21495: 21489: 21479: 21478: 21477: 21461: 21458: 21448: 21436: 21414: 21382: 21379: 21369: 21362: 21359: 21356: 21347: 21341: 21339: 21330: 21324: 21314: 21305: 21301: 21297: 21294: 21289: 21282: 21274: 21263: 21259: 21254: 21248: 21244: 21240: 21238: 21224: 21218: 21214: 21209: 21199: 21192: 21182: 21176: 21172: 21156: 21149: 21141: 21122: 21116: 21112: 21108: 21106: 21095: 21081: 21080: 21079: 21062: 21057: 21047: 21038: 21034: 21028: 21024: 21020: 21011: 21001: 21000: 20999: 20983: 20978: 20974: 20968: 20964: 20958: 20954: 20947: 20944: 20940: 20911: 20906: 20901: 20894: 20891: 20882: 20878: 20872: 20868: 20864: 20862: 20853: 20847: 20837: 20830: 20825: 20815: 20807: 20801: 20798: 20794: 20788: 20780: 20775: 20772: 20769: 20765: 20761: 20756: 20751: 20744: 20741: 20732: 20728: 20722: 20718: 20714: 20712: 20692: 20688: 20676: 20675: 20674: 20657: 20652: 20647: 20643: 20639: 20636: 20632: 20625: 20617: 20612: 20608: 20603: 20597: 20593: 20585: 20584: 20583: 20563: 20551: 20537: 20510: 20507: 20504: 20500: 20495: 20489: 20479: 20472: 20467: 20457: 20449: 20443: 20440: 20436: 20430: 20427: 20424: 20420: 20416: 20411: 20406: 20401: 20394: 20386: 20377: 20370: 20365: 20361: 20353: 20348: 20345: 20342: 20338: 20333: 20329: 20326: 20324: 20316: 20313: 20309: 20302: 20293: 20285: 20277: 20271: 20259: 20252: 20241: 20238: 20236: 20230: 20221: 20215: 20196: 20195: 20194: 20186: 20184: 20165: 20162: 20156: 20151: 20147: 20135: 20128: 20125: 20113: 20102: 20097: 20094: 20091: 20087: 20079: 20078: 20077: 20074: 20060: 20057: 20054: 20051: 20047: 20043: 20040: 20013: 20007: 20001: 19998: 19993: 19988: 19981: 19978: 19969: 19965: 19959: 19956: 19951: 19949: 19940: 19933: 19927: 19924: 19919: 19914: 19907: 19904: 19895: 19891: 19885: 19882: 19876: 19872: 19867: 19862: 19855: 19852: 19843: 19839: 19835: 19832: 19830: 19822: 19819: 19811: 19807: 19800: 19792: 19785: 19782: 19770: 19759: 19754: 19751: 19748: 19744: 19740: 19738: 19733: 19722: 19721: 19720: 19703: 19697: 19691: 19688: 19683: 19678: 19671: 19668: 19659: 19655: 19649: 19646: 19641: 19638: 19631: 19630: 19629: 19610: 19603: 19600: 19588: 19584: 19572: 19566: 19555: 19551: 19547: 19542: 19535: 19532: 19523: 19516: 19513: 19504: 19500: 19496: 19493: 19489: 19482: 19476: 19473: 19468: 19463: 19456: 19453: 19444: 19440: 19434: 19431: 19425: 19418: 19415: 19411: 19406: 19400: 19394: 19387: 19386: 19385: 19383: 19379: 19360: 19351: 19348: 19342: 19333: 19327: 19321: 19313: 19309: 19305: 19302: 19298: 19294: 19271: 19268: 19265: 19260: 19250: 19244: 19238: 19235: 19230: 19225: 19218: 19215: 19206: 19201: 19198: 19195: 19191: 19185: 19182: 19176: 19172: 19169: 19167: 19159: 19156: 19151: 19144: 19135: 19132: 19126: 19120: 19114: 19110: 19106: 19103: 19100: 19098: 19090: 19069: 19068: 19067: 19065: 19061: 19057: 19053: 19049: 19046:, coordinate 19045: 19016: 19013: 19009: 19005: 19001: 18997: 18990: 18984: 18980: 18963: 18958: 18948: 18927: 18922: 18916: 18908: 18902: 18896: 18891: 18882: 18865: 18860: 18850: 18829: 18825: 18822: 18816: 18810: 18806: 18789: 18784: 18763: 18759: 18754: 18750: 18745: 18739: 18727: 18726: 18725: 18722: 18708: 18681: 18676: 18672: 18666: 18659: 18653: 18649: 18632: 18627: 18606: 18602: 18597: 18593: 18588: 18582: 18574: 18571: 18568: 18566: 18558: 18553: 18549: 18543: 18536: 18530: 18526: 18511: 18507: 18502: 18493: 18479: 18475: 18470: 18462: 18446: 18443: 18439: 18435: 18432: 18430: 18422: 18417: 18413: 18408: 18398: 18392: 18389: 18381: 18372: 18355: 18350: 18329: 18325: 18319: 18313: 18310: 18306: 18292: 18279: 18274: 18270: 18267: 18265: 18257: 18252: 18248: 18228: 18222: 18219: 18216: 18214: 18193: 18187: 18176: 18175: 18174: 18155: 18151: 18145: 18137: 18121: 18114: 18113: 18112: 18098: 18089: 18087: 18083: 18079: 18075: 18071: 18067: 18063: 18059: 18055: 18051: 18047: 18043: 18039: 18035: 18031: 18030:test function 18027: 18023: 18019: 18015: 18011: 18001: 17998: 17985: 17980: 17975: 17971: 17967: 17955: 17947: 17943: 17934: 17930: 17926: 17917: 17909: 17905: 17896: 17892: 17884: 17867: 17862: 17841: 17837: 17832: 17827: 17823: 17811: 17808: − 17804: 17801: = 17797: 17779: 17774: 17770: 17764: 17756: 17735: 17731: 17722: 17718: 17714: 17693: 17689: 17680: 17676: 17672: 17651: 17647: 17643: 17638: 17634: 17621: 17606: 17601: 17597: 17591: 17583: 17565: 17561: 17538: 17533: 17529: 17523: 17515: 17497: 17493: 17481: 17474: 17464: 17462: 17458: 17442: 17439: 17435: 17424: 17415: 17410: 17406: 17402: 17398: 17374: 17371: 17366: 17362: 17358: 17353: 17349: 17343: 17336: 17328: 17327: 17326: 17323: 17313: 17311: 17307: 17303: 17299: 17295: 17291: 17272: 17267: 17263: 17259: 17253: 17247: 17238: 17233: 17203: 17197: 17193: 17185: 17184: 17183: 17167: 17163: 17154: 17135: 17132: 17128: 17122: 17118: 17114: 17108: 17102: 17093: 17088: 17058: 17052: 17040: 17039: 17038: 17036: 17013: 17008: 17004: 16992: 16988: 16982: 16975: 16971: 16969: 16959: 16955: 16942: 16936: 16927: 16922: 16891: 16884: 16880: 16877: 16872: 16858: 16852: 16848: 16838: 16833: 16802: 16794: 16788: 16766: 16760: 16756: 16752: 16750: 16744: 16740: 16735: 16715: 16711: 16706: 16702: 16691: 16690: 16689: 16687: 16683: 16679: 16676:Now, for any 16674: 16672: 16669: 16666: 16665:one parameter 16662: 16658: 16654: 16650: 16646: 16642: 16623: 16617: 16614: 16608: 16602: 16597: 16589: 16583: 16577: 16564: 16558: 16541: 16540: 16539: 16537: 16515: 16507: 16503: 16497: 16489: 16480: 16464: 16457: 16456: 16455: 16453: 16432: 16428: 16424: 16417: 16414: 16411: 16402: 16399: 16393: 16387: 16381: 16373: 16369: 16363: 16356: 16352: 16348: 16344: 16339: 16319: 16315: 16310: 16306: 16299: 16298: 16297: 16295: 16291: 16288: 16260: 16255: 16253: 16249: 16233: 16210: 16207: 16198: 16192: 16189: 16181: 16168: 16158: 16157: 16156: 16142: 16110: 16094: 16062: 16058: 16042: 16034: 16030: 16026: 16010: 16002: 15983: 15980: 15975: 15971: 15963: 15960: 15954: 15948: 15943: 15935: 15929: 15923: 15908: 15904: 15899: 15892: 15875: 15874: 15873: 15836: 15828: 15812: 15804: 15782: 15779: 15776: 15771: 15763: 15760: 15743: 15742: 15741: 15739: 15735: 15716: 15698: 15693: 15647: 15628: 15610: 15596: 15595: 15594: 15593: 15571: 15556: 15538: 15509: 15505: 15501: 15498: 15495: 15490: 15486: 15477: 15476:scalar fields 15473: 15470: 15466: 15462: 15458: 15454: 15451: 15447: 15421: 15418:formulation, 15417: 15413: 15409: 15408: 15407: 15405: 15400: 15398: 15394: 15390: 15386: 15382: 15378: 15350: 15346: 15342: 15338: 15314: 15308: 15299: 15291: 15288: 15281: 15277: 15266: 15257: 15253: 15249: 15243: 15239: 15234: 15231: 15226: 15222: 15216: 15199: 15196: 15189: 15185: 15174: 15164: 15160: 15155: 15151: 15143: 15142: 15141: 15124: 15121: 15116: 15112: 15103: 15099: 15082: 15081: 15080: 15078: 15074: 15055: 15049: 15045: 15039: 15027: 15024: 15019: 15011: 15008: 15003: 14999: 14989: 14979: 14978: 14977: 14960: 14954: 14950: 14941: 14937: 14926: 14922: 14912: 14907: 14903: 14897: 14881: 14880: 14879: 14865: 14862: 14857: 14854: 14847: 14843: 14818: 14814: 14805: 14787: 14783: 14774: 14756: 14752: 14746: 14708: 14704: 14698: 14686: 14683: 14678: 14674: 14664: 14658: 14653: 14645: 14642: 14640: 14633: 14629: 14625: 14616: 14612: 14608: 14605: 14603: 14596: 14592: 14588: 14577: 14576: 14575: 14573: 14554: 14550: 14547: 14543: 14537: 14533: 14529: 14525: 14519: 14515: 14511: 14506: 14503: 14496: 14492: 14486: 14481: 14477: 14472: 14468: 14465: 14460: 14456: 14446: 14435: 14432: 14425: 14421: 14410: 14400: 14391: 14387: 14370: 14369: 14368: 14351: 14347: 14344: 14341: 14336: 14332: 14327: 14321: 14317: 14313: 14309: 14303: 14299: 14295: 14290: 14287: 14280: 14276: 14270: 14265: 14261: 14256: 14252: 14249: 14244: 14240: 14230: 14219: 14216: 14209: 14205: 14194: 14184: 14175: 14171: 14152: 14144: 14143: 14142: 14121: 14117: 14111: 14107: 14097: 14086: 14083: 14076: 14072: 14061: 14051: 14042: 14038: 14025: 14020: 14017: 14010: 14006: 13994: 13983: 13980: 13973: 13969: 13958: 13949: 13944: 13940: 13930: 13923: 13915: 13912: 13905: 13901: 13890: 13881: 13872: 13868: 13851: 13843: 13838: 13832: 13828: 13824: 13819: 13816: 13809: 13805: 13799: 13794: 13790: 13785: 13781: 13778: 13774: 13768: 13764: 13760: 13755: 13752: 13745: 13741: 13735: 13730: 13726: 13721: 13717: 13713: 13700: 13699: 13698: 13676: 13672: 13663: 13654: 13650: 13642: 13639: 13632: 13628: 13617: 13608: 13599: 13595: 13578: 13577: 13576: 13559: 13554: 13548: 13544: 13534: 13527: 13518: 13514: 13501: 13493: 13489: 13478: 13474: 13458: 13452: 13447: 13444: 13437: 13433: 13421: 13411: 13410: 13409: 13408: 13389: 13383: 13380: 13373: 13369: 13357: 13346: 13343: 13336: 13332: 13321: 13312: 13307: 13303: 13293: 13282: 13278: 13269: 13260: 13256: 13251: 13245: 13241: 13237: 13232: 13229: 13222: 13218: 13212: 13207: 13203: 13198: 13194: 13191: 13187: 13181: 13177: 13173: 13168: 13165: 13158: 13154: 13148: 13143: 13139: 13134: 13130: 13126: 13118: 13117: 13116: 13099: 13095: 13092: 13089: 13084: 13080: 13075: 13070: 13064: 13060: 13056: 13052: 13046: 13042: 13038: 13033: 13030: 13023: 13019: 13013: 13008: 13004: 12999: 12995: 12991: 12982: 12978: 12965: 12961: 12956: 12950: 12946: 12942: 12937: 12934: 12927: 12923: 12917: 12912: 12908: 12903: 12899: 12896: 12892: 12886: 12882: 12878: 12873: 12870: 12863: 12859: 12853: 12848: 12844: 12839: 12835: 12831: 12826: 12816: 12808: 12807: 12806: 12804: 12785: 12776: 12772: 12761: 12757: 12747: 12742: 12739: 12732: 12728: 12718: 12717: 12716: 12699: 12696: 12693: 12688: 12684: 12679: 12673: 12669: 12665: 12660: 12657: 12650: 12646: 12640: 12635: 12631: 12626: 12622: 12613: 12609: 12606: 12601: 12597: 12592: 12586: 12582: 12578: 12573: 12570: 12563: 12559: 12553: 12548: 12544: 12539: 12535: 12519: 12511: 12510: 12509: 12507: 12502: 12484: 12479: 12474: 12470: 12466: 12460: 12456: 12446: 12440: 12436: 12431: 12427: 12423: 12417: 12413: 12409: 12405: 12400: 12396: 12392: 12386: 12382: 12374: 12373: 12372: 12370: 12366: 12348: 12344: 12340: 12317: 12312: 12307: 12303: 12299: 12293: 12289: 12285: 12282: 12278: 12273: 12269: 12265: 12259: 12255: 12251: 12247: 12242: 12238: 12234: 12228: 12224: 12215: 12211: 12203: 12202: 12201: 12182: 12178: 12174: 12171: 12166: 12162: 12158: 12153: 12149: 12140: 12136: 12128: 12127: 12126: 12124: 12120: 12116: 12112: 12108: 12090: 12086: 12071: 12054: 12048: 12040: 12025: 12009: 12000: 11997: 11993: 11989: 11986: 11980: 11963: 11947: 11937: 11929: 11928: 11927: 11913: 11910: 11894: 11880: 11858: 11850: 11835: 11819: 11810: 11807: 11803: 11799: 11796: 11790: 11766: 11751: 11735: 11725: 11717: 11716: 11715: 11694: 11686: 11682: 11672: 11664: 11649: 11633: 11624: 11616: 11612: 11602: 11594: 11579: 11563: 11554: 11550: 11542: 11534: 11530: 11520: 11496: 11481: 11465: 11456: 11453: 11445: 11441: 11431: 11407: 11392: 11376: 11367: 11364: 11356: 11352: 11348: 11340: 11336: 11326: 11315: 11307: 11303: 11288: 11285: 11282: 11274: 11270: 11266: 11258: 11254: 11244: 11233: 11225: 11221: 11206: 11199: 11196: 11185: 11184: 11183: 11166: 11160: 11138: 11130: 11125: 11108: 11092: 11083: 11077: 11069: 11047: 11038: 11032: 11010: 11002: 10997: 10980: 10964: 10955: 10949: 10941: 10931: 10921: 10905: 10893: 10890: 10886: 10880: 10876: 10872: 10865: 10857: 10842: 10826: 10816: 10809: 10806: 10802: 10793: 10792: 10791: 10770: 10767: 10764: 10757: 10735: 10727: 10722: 10705: 10689: 10680: 10674: 10666: 10644: 10631: 10627: 10619: 10615: 10610: 10606: 10602: 10594: 10586: 10582: 10572: 10548: 10533: 10517: 10508: 10505: 10497: 10493: 10483: 10459: 10444: 10428: 10419: 10416: 10408: 10404: 10400: 10392: 10388: 10378: 10367: 10359: 10355: 10340: 10337: 10334: 10326: 10322: 10318: 10310: 10306: 10296: 10285: 10277: 10273: 10258: 10251: 10245: 10236: 10233: 10227: 10224: 10220: 10214: 10211: 10200: 10199: 10198: 10177: 10174: 10167: 10143: 10128: 10112: 10103: 10100: 10094: 10082: 10075: 10059: 10038: 10033: 10021: 10011: 9995: 9986: 9983: 9977: 9953: 9931: 9922: 9920: 9912: 9905: 9881: 9866: 9850: 9841: 9838: 9832: 9820: 9803: 9791: 9788: 9784: 9778: 9768: 9752: 9743: 9740: 9734: 9710: 9700: 9690: 9674: 9662: 9659: 9655: 9649: 9645: 9643: 9637: 9633: 9627: 9603: 9588: 9572: 9562: 9555: 9552: 9548: 9535: 9534: 9533: 9512: 9509: 9506: 9501: 9497: 9491: 9467: 9458: 9452: 9430: 9422: 9417: 9406: 9403: 9398: 9391: 9374: 9353: 9348: 9337: 9333: 9323: 9307: 9298: 9294: 9287: 9279: 9270: 9267: 9261: 9237: 9228: 9224: 9207: 9194: 9190: 9182: 9178: 9173: 9169: 9165: 9157: 9149: 9145: 9141: 9133: 9129: 9119: 9108: 9100: 9096: 9081: 9078: 9075: 9067: 9063: 9059: 9051: 9047: 9037: 9026: 9018: 9014: 8999: 8992: 8986: 8977: 8974: 8968: 8965: 8961: 8955: 8952: 8941: 8940: 8939: 8937: 8933: 8929: 8905: 8902: 8898: 8890: 8887: 8883: 8877: 8874: 8871: 8867: 8864: 8854: 8840: 8837: 8831: 8828: 8825: 8821: 8818: 8790: 8781: 8775: 8772: 8766: 8763: 8760: 8756: 8753: 8738: 8732: 8727: 8724: 8721: 8716: 8712: 8706: 8703: 8700: 8695: 8691: 8686: 8682: 8680: 8671: 8668: 8664: 8656: 8653: 8649: 8642: 8639: 8631: 8625: 8613: 8606: 8603: 8595: 8583: 8578: 8575: 8572: 8567: 8563: 8557: 8554: 8551: 8546: 8542: 8537: 8533: 8531: 8523: 8516: 8513: 8501: 8500: 8499: 8482: 8476: 8473: 8470: 8466: 8463: 8453: 8439: 8436: 8430: 8427: 8424: 8420: 8417: 8389: 8380: 8374: 8371: 8365: 8351: 8345: 8342: 8338: 8333: 8326: 8323: 8315: 8309: 8291: 8282: 8267: 8266: 8265: 8263: 8259: 8233: 8230: 8224: 8221: 8218: 8214: 8211: 8196: 8193: 8187: 8184: 8178: 8164: 8161: 8154: 8151: 8143: 8132: 8124: 8109: 8106: 8103: 8100: 8097: 8093: 8090: 8084: 8079: 8068: 8067: 8066: 8064: 8060: 8041: 8035: 8016: 8007: 8001: 7989: 7973: 7961: 7958: 7954: 7945: 7944: 7943: 7942: 7937: 7924: 7921: 7914: 7911: 7905: 7896: 7885: 7879: 7865: 7858: 7854: 7846: 7842: 7837: 7833: 7830: 7822: 7820: 7805: 7803: 7799: 7780: 7772: 7769: 7765: 7760: 7756: 7748: 7744: 7733: 7729: 7719: 7714: 7710: 7701: 7697: 7688: 7678: 7674: 7671: 7666: 7662: 7654: 7653: 7652: 7650: 7631: 7626: 7622: 7618: 7613: 7609: 7605: 7600: 7597: 7593: 7587: 7583: 7577: 7569: 7564: 7556: 7553: 7546: 7545: 7544: 7542: 7538: 7534: 7530: 7526: 7522: 7519:are equal to 7518: 7514: 7510: 7506: 7487: 7479: 7475: 7469: 7466: 7463: 7459: 7450: 7446: 7439: 7433: 7428: 7425: 7421: 7414: 7407: 7406: 7405: 7403: 7399: 7395: 7391: 7387: 7383: 7379: 7375: 7371: 7367: 7363: 7360: 7356: 7352: 7348: 7344: 7339: 7337: 7321: 7301: 7281: 7258: 7255: 7250: 7247: 7238: 7229: 7225: 7217: 7216: 7215: 7189: 7184: 7180: 7176: 7171: 7168: 7164: 7160: 7156: 7148: 7145: 7141: 7119: 7115: 7107: 7104: 7100: 7078: 7073: 7063: 7058: 7054: 7050: 7038: 7033: 7029: 7025: 7022: 7017: 7008: 7004: 6996: 6995: 6994: 6980: 6960: 6940: 6920: 6911: 6907: 6904: 6876: 6871: 6867: 6863: 6860: 6855: 6850: 6838: 6837: 6836: 6821: 6815: 6810: 6806: 6800: 6796: 6792: 6787: 6783: 6778: 6763: 6758: 6754: 6750: 6719: 6716: 6711: 6701: 6696: 6692: 6688: 6685: 6680: 6668: 6667: 6666: 6650: 6646: 6623: 6619: 6615: 6612: 6607: 6603: 6580: 6576: 6554: 6548: 6543: 6539: 6533: 6529: 6525: 6520: 6516: 6511: 6507: 6496: 6492: 6485: 6477: 6474:denoting the 6461: 6439: 6434: 6430: 6424: 6420: 6416: 6411: 6407: 6398: 6394: 6385: 6368: 6362: 6358: 6354: 6344: 6336: 6327: 6323: 6314: 6312: 6293: 6290: 6287: 6282: 6278: 6272: 6260: 6259: 6258: 6256: 6251: 6237: 6217: 6209: 6205: 6184: 6176: 6168: 6165: 6161: 6139: 6134: 6129: 6121: 6116: 6111: 6107: 6099: 6098: 6097: 6096: 6080: 6057: 6052: 6047: 6037: 6027: 6023: 6019: 5995: 5994: 5993: 5976: 5971: 5961: 5957: 5953: 5950: 5944: 5937: 5936: 5935: 5921: 5918: 5915: 5912: 5909: 5906: 5903: 5900: 5897: 5894: 5886: 5867: 5862: 5852: 5844: 5841: 5817: 5816: 5815: 5733: 5711: 5707: 5663: 5657: 5654: 5651: 5645: 5638: 5637: 5636: 5622: 5613: 5611: 5607: 5588: 5583: 5579: 5573: 5567: 5563: 5559: 5556: 5551: 5543: 5540: 5536: 5525: 5522: 5508: 5507: 5506: 5504: 5485: 5482: 5471: 5451: 5444: 5439: 5437: 5433: 5422: 5420: 5416: 5412: 5408: 5404: 5385: 5377: 5369: 5365: 5356: 5347: 5343: 5335: 5331: 5322: 5313: 5309: 5301: 5293: 5284: 5268: 5255: 5254: 5253: 5236: 5228: 5220: 5208: 5207: 5206: 5204: 5200: 5196: 5192: 5188: 5184: 5180: 5161: 5153: 5144: 5141: 5138: 5118: 5117: 5116: 5114: 5110: 5106: 5101: 5097: 5093: 5090: 5085: 5084: 5080: 5078: 5074: 5069: 5065: 5061: 5057: 5038: 5029: 5024: 5020: 5008: 4999: 4994: 4990: 4982: 4981: 4980: 4978: 4974: 4971: 4966: 4962: 4958: 4954: 4949: 4945: 4940: 4936: 4931: 4927: 4922: 4918: 4913: 4912: 4894: 4891: 4888: 4882: 4871: 4862: 4846: 4837: 4834: 4827: 4826: 4825: 4824: 4821: 4817: 4813: 4809: 4805: 4801: 4797: 4793: 4788: 4787: 4778: 4776: 4755: 4745: 4736: 4720: 4711: 4706: 4702: 4697: 4693: 4690: 4684: 4673: 4664: 4648: 4638: 4630: 4629: 4628: 4626: 4622: 4617: 4615: 4611: 4607: 4602: 4598: 4589: 4585: 4582: 4579: 4576: 4572: 4566: 4563: 4562: 4561: 4559: 4558:infinitesimal 4554: 4550: 4527: 4519: 4507: 4503: 4497: 4493: 4489: 4487: 4477: 4468: 4464: 4458: 4454: 4448: 4444: 4440: 4438: 4433: 4430: 4419: 4418: 4417: 4414: 4412: 4408: 4404: 4400: 4396: 4393: 4389: 4366: 4355: 4352: 4344: 4327: 4313: 4310: 4307: 4304: 4301: 4292: 4286: 4281: 4270: 4269: 4268: 4266: 4263: 4259: 4255: 4250: 4241: 4231: 4229: 4225: 4220: 4218: 4213: 4209: 4204: 4202: 4198: 4191: 4184: 4179: 4157: 4150: 4147: 4135: 4126: 4121: 4117: 4109: 4108: 4107: 4090: 4084: 4081: 4075: 4072: 4065: 4061: 4057: 4051: 4047: 4039: 4032: 4029: 4017: 4008: 4001: 3998: 3994: 3985: 3984: 3983: 3981: 3959: 3940: 3931: 3927: 3918: 3902: 3893: 3886: 3883: 3879: 3870: 3869: 3868: 3866: 3862: 3858: 3854: 3850: 3832: 3823: 3820: 3810: 3804: 3798: 3783: 3782: 3781: 3779: 3775: 3756: 3750: 3747: 3740: 3737: 3731: 3720: 3709: 3706: 3703: 3700: 3693: 3692: 3691: 3690: 3686: 3682: 3678: 3674: 3670: 3667: 3663: 3659: 3654: 3652: 3648: 3644: 3640: 3636: 3632: 3628: 3624: 3620: 3616: 3612: 3608: 3604: 3600: 3595: 3592: 3588: 3569: 3563: 3560: 3554: 3551: 3545: 3539: 3536: 3531: 3528: 3518: 3517: 3516: 3514: 3510: 3506: 3502: 3498: 3494: 3488: 3484: 3480: 3454: 3429: 3425: 3404: 3381: 3353: 3349: 3341: 3325: 3314: 3299: 3274: 3270: 3247: 3243: 3234: 3216: 3213: 3208: 3204: 3180: 3155: 3151: 3147: 3142: 3138: 3127:the interval 3126: 3125: 3108: 3105: 3100: 3096: 3089: 3066: 3060: 3052: 3036: 3030: 3023: 3020: 3012: 3006: 2999: 2996: 2985: 2981: 2974: 2968: 2964: 2960: 2957: 2953: 2949: 2940: 2927: 2923: 2917: 2910: 2907: 2895: 2888: 2885: 2873: 2864: 2861: 2857: 2853: 2850: 2847: 2843: 2837: 2830: 2827: 2812: 2805: 2802: 2790: 2781: 2778: 2775: 2772: 2768: 2764: 2761: 2758: 2734: 2711: 2708: 2705: 2699: 2691: 2674: 2670: 2665: 2659: 2652: 2649: 2638: 2634: 2627: 2621: 2617: 2594: 2590: 2586: 2581: 2577: 2568: 2564: 2541: 2537: 2528: 2527: 2525: 2522: 2508: 2505: 2501: 2494: 2491: 2481: 2477: 2470: 2449: 2429: 2425: 2418: 2415: 2405: 2401: 2394: 2373: 2365: 2361: 2353: 2349: 2340: 2337: 2326: 2316: 2312: 2304: 2300: 2292: 2288: 2279: 2276: 2265: 2255: 2234: 2209: 2205: 2182: 2178: 2168: 2155: 2152: 2143: 2140: 2129: 2120: 2117: 2114: 2111: 2102: 2099: 2088: 2079: 2073: 2069: 2063: 2059: 2056: 2050: 2046: 2036: 2033: 2022: 2013: 2010: 2004: 2001: 1986: 1983: 1972: 1963: 1960: 1957: 1951: 1948: 1945: 1919: 1916: 1897: 1894: 1884: 1880: 1867: 1864: 1852: 1838: 1834: 1830: 1827: 1824: 1818: 1815: 1803: 1800: 1777: 1774: 1771: 1768: 1764: 1760: 1757: 1737: 1717: 1714: 1694: 1690: 1686: 1683: 1660: 1657: 1631: 1628: 1618:and velocity 1605: 1585: 1582: 1579: 1576: 1556: 1536: 1516: 1488: 1484: 1480: 1475: 1471: 1447: 1438: 1424: 1411: 1404: 1386: 1378: 1374: 1358: 1351: 1347: 1328: 1322: 1313: 1299: 1296: 1293: 1290: 1284: 1281: 1278: 1258: 1245: 1235: 1233: 1229: 1224: 1220: 1214: 1212: 1208: 1203: 1201: 1197: 1193: 1189: 1183: 1181: 1177: 1173: 1169: 1165: 1159: 1157: 1153: 1147: 1143: 1139: 1131: 1129: 1125: 1120: 1118: 1114: 1108: 1106: 1102: 1098: 1092: 1089: 1085: 1081: 1071: 1069: 1066:systems with 1065: 1061: 1057: 1053: 1049: 1045: 1040: 1038: 1034: 1030: 1026: 1022: 1018: 1014: 1010: 1006: 1002: 998: 987: 982: 980: 975: 973: 968: 967: 965: 964: 957: 954: 952: 949: 947: 944: 942: 939: 937: 934: 932: 929: 927: 924: 923: 915: 914: 907: 904: 902: 899: 897: 894: 892: 889: 888: 880: 879: 868: 865: 863: 860: 858: 855: 854: 853: 852: 842: 841: 830: 827: 825: 822: 820: 817: 815: 812: 810: 809:Line integral 807: 805: 802: 800: 797: 796: 795: 794: 790: 789: 784: 781: 779: 776: 774: 771: 769: 766: 765: 764: 763: 759: 758: 752: 751:Multivariable 746: 745: 734: 731: 729: 726: 724: 721: 719: 716: 714: 711: 709: 706: 705: 704: 703: 699: 698: 693: 690: 688: 685: 683: 680: 678: 675: 673: 670: 668: 665: 664: 663: 662: 656: 650: 649: 638: 635: 633: 630: 628: 625: 623: 620: 618: 614: 612: 609: 607: 604: 602: 599: 597: 594: 592: 589: 588: 587: 586: 583: 580: 579: 574: 571: 569: 566: 564: 561: 559: 556: 554: 551: 548: 544: 541: 540: 539: 538: 532: 526: 525: 514: 511: 509: 506: 504: 501: 499: 496: 494: 491: 489: 486: 483: 479: 475: 474:trigonometric 471: 468: 466: 463: 461: 458: 456: 453: 452: 451: 450: 446: 445: 440: 437: 435: 432: 430: 427: 425: 422: 419: 415: 412: 410: 407: 406: 405: 404: 400: 399: 394: 391: 389: 386: 384: 381: 380: 379: 378: 372: 366: 365: 354: 351: 349: 346: 344: 341: 339: 336: 334: 331: 329: 326: 324: 321: 319: 316: 314: 311: 309: 306: 305: 304: 303: 300: 297: 296: 291: 288: 286: 285:Related rates 283: 281: 278: 276: 273: 271: 268: 266: 263: 262: 261: 260: 256: 255: 248: 245: 243: 242:of a function 240: 238: 237:infinitesimal 235: 234: 233: 230: 227: 223: 220: 219: 218: 217: 213: 212: 206: 200: 199: 193: 190: 188: 185: 183: 180: 179: 174: 171: 169: 166: 165: 161: 158: 157: 156: 155: 136: 130: 127: 121: 115: 112: 109: 106: 99: 92: 89: 83: 78: 74: 65: 64: 61: 58: 57: 53: 52: 46: 41: 37: 33: 19: 23762: 23756: 23722: 23711:. Retrieved 23704:the original 23684:(1): 29–36. 23681: 23677: 23624: 23620: 23591: 23587: 23575:. Retrieved 23572:math.ucr.edu 23571: 23542:Gott. Nachr. 23541: 23501: 23497: 23491:(in German). 23458: 23428: 23425:Olver, Peter 23413:. Retrieved 23408: 23377: 23362:. Retrieved 23354: 23307: 23284:cite journal 23233: 23227: 23217: 23197: 23190: 23169: 23136: 23094: 23087: 23059: 23052: 23039: 23019: 23012: 22979: 22975: 22965: 22938: 22932: 22904: 22897: 22869: 22850: 22846: 22836: 22742: 22722:phase factor 22707: 22664: 22661:Applications 22648: 22602: 22341: 22285: 22130: 21919: 21820: 21818: 21594: 21434: 21403: 21077: 20930: 20672: 20552: 20529: 20192: 20181:(called the 20180: 20075: 20032: 19718: 19627: 19381: 19377: 19311: 19307: 19303: 19290: 19063: 19055: 19051: 19047: 19043: 19041: 18723: 18700: 18172: 18090: 18085: 18082:distribution 18081: 18077: 18073: 18069: 18065: 18061: 18057: 18053: 18049: 18045: 18041: 18037: 18033: 18025: 18021: 18017: 18013: 18009: 18007: 17999: 17809: 17802: 17795: 17622: 17479: 17472: 17470: 17389: 17319: 17309: 17289: 17287: 17182:defined by: 17150: 17034: 17032: 16677: 16675: 16660: 16656: 16648: 16638: 16535: 16533: 16451: 16449: 16293: 16256: 16225: 16056: 16028: 15998: 15800: 15737: 15699:. We assume 15694: 15643: 15589: 15468: 15460: 15457:field theory 15419: 15403: 15401: 15396: 15393:fiber bundle 15388: 15384: 15348: 15344: 15336: 15334: 15139: 15076: 15072: 15070: 14975: 14803: 14729: 14569: 14366: 14140: 13696: 13574: 13404: 13114: 12800: 12714: 12503: 12499: 12368: 12364: 12332: 12199: 12122: 12118: 12114: 12110: 12106: 12077: 12069: 11878: 11876: 11713: 11181: 10789: 10196: 9531: 8931: 8927: 8925: 8497: 8261: 8257: 8255: 8062: 8056: 7938: 7823: 7818: 7816: 7804:of physics. 7798:Hermann Weyl 7795: 7648: 7646: 7524: 7520: 7516: 7515:, while the 7512: 7508: 7504: 7502: 7401: 7400:that leave | 7397: 7389: 7385: 7381: 7373: 7369: 7365: 7361: 7350: 7346: 7340: 7273: 7213: 6933:in place of 6909: 6905: 6900: 6734: 6383: 6315: 6308: 6252: 6207: 6201: 6072: 5991: 5882: 5678: 5614: 5605: 5603: 5440: 5428: 5414: 5410: 5406: 5402: 5400: 5251: 5202: 5198: 5194: 5190: 5186: 5185:= 1. Taking 5182: 5178: 5176: 5108: 5104: 5103:of an angle 5099: 5095: 5091: 5086: 5082: 5081: 5072: 5067: 5063: 5053: 4976: 4972: 4964: 4960: 4956: 4952: 4947: 4943: 4938: 4934: 4929: 4925: 4920: 4916: 4914: 4910: 4909: 4822: 4815: 4811: 4807: 4803: 4799: 4795: 4791: 4789: 4785: 4784: 4772: 4624: 4618: 4609: 4600: 4596: 4594: 4587: 4583: 4567: 4552: 4548: 4546: 4415: 4410: 4406: 4398: 4394: 4391: 4387: 4385: 4264: 4257: 4253: 4251: 4248: 4221: 4205: 4193: 4186: 4182: 4180: 4177: 4105: 3976: 3974: 3860: 3856: 3852: 3847: 3777: 3773: 3771: 3688: 3680: 3668: 3655: 3619:Isaac Newton 3596: 3590: 3584: 3504: 3496: 3490: 3444:through the 2748: 2523: 2169: 1853: 1647:change, but 1439: 1409: 1402: 1314: 1250: 1231: 1227: 1222: 1218: 1216: 1210: 1204: 1191: 1187: 1184: 1161: 1149: 1145: 1141: 1137: 1130:also exist. 1121: 1116: 1112: 1109: 1093: 1077: 1041: 1021:Emmy Noether 996: 995: 470:Substitution 232:Differential 205:Differential 45:Emmy Noether 36: 23397:Online copy 23133:Lanczos, C. 22672:translation 20183:Hamiltonian 18028: is a 18016:satisfying 15801:called the 15644:called the 15416:Hamiltonian 7808:Derivations 6204:dot product 6202:(where the 5610:jet bundles 4627:quantities 4224:Felix Klein 4185:coordinate 3515:) is zero, 1676:changes by 1207:Felix Klein 1128:superspaces 1064:dissipative 926:Precalculus 919:Miscellanea 884:Specialized 791:Definitions 558:Alternating 401:Definitions 214:Definitions 23800:Categories 23747:1902.01989 23713:2014-11-12 23655:1804.01714 23564:Baez, John 22853:: 235–257. 22829:References 22704:velocity). 18052:satisfies 17455:) probing 17302:space-like 17037:, we have 16296:such that 16290:derivation 16287:functional 15827:derivative 15592:functional 15555:isomorphic 6093:conserved 5432:space–time 5405:along the 4238:See also: 3685:Lagrangian 3639:divergence 3509:derivative 1373:stationary 1200:solenoidal 1164:covariance 1084:Lagrangian 1029:Lagrangian 906:Variations 901:Stochastic 891:Fractional 760:Formalisms 723:Divergence 692:Identities 672:Divergence 222:Derivative 173:Continuity 23772:0906.1732 23577:28 August 23536:119019843 23415:28 August 23364:28 August 23342:125902566 23004:0217-7323 22957:759172774 22889:857769535 22655:anomalies 22626:μ 22616:μ 22612:∂ 22570:φ 22566:λ 22563:− 22560:φ 22555:ν 22551:∂ 22547:φ 22542:ν 22538:∂ 22517:μ 22509:− 22501:φ 22495:φ 22490:ν 22486:∂ 22480:ν 22467:φ 22462:μ 22458:∂ 22442:μ 22434:− 22428:φ 22401:φ 22396:μ 22392:∂ 22385:∂ 22381:∂ 22362:μ 22315:μ 22302:μ 22257:μ 22242:μ 22238:∂ 22220:φ 22214:μ 22206:λ 22203:− 22200:φ 22195:ν 22191:∂ 22187:φ 22182:ν 22178:∂ 22172:μ 22147:μ 22143:∂ 22108:φ 22102:φ 22097:μ 22093:∂ 22087:μ 22068:φ 22064:λ 22058:− 22050:φ 22045:μ 22041:∂ 22034:φ 22029:ν 22025:∂ 22019:μ 22015:∂ 22009:ν 21998:φ 21993:μ 21989:∂ 21980:φ 21975:μ 21971:∂ 21928:φ 21893:φ 21878:φ 21873:μ 21869:∂ 21863:μ 21840:φ 21776:φ 21772:λ 21769:− 21766:φ 21761:μ 21757:∂ 21753:φ 21748:μ 21744:∂ 21725:∫ 21685:φ 21680:μ 21676:∂ 21660:φ 21645:∫ 21632:φ 21557:˙ 21552:→ 21537:− 21531:→ 21522:⇒ 21499:→ 21452:→ 21418:→ 21373:→ 21360:− 21351:→ 21325:α 21318:→ 21306:α 21298:− 21290:α 21283:˙ 21278:→ 21264:α 21249:α 21245:∑ 21228:→ 21219:− 21210:α 21203:→ 21186:→ 21177:⋅ 21157:α 21150:˙ 21145:→ 21133:∂ 21129:∂ 21117:α 21113:∑ 21099:→ 21058:α 21051:→ 21039:α 21029:α 21025:∑ 21015:→ 20979:α 20969:α 20959:α 20955:∑ 20902:α 20895:˙ 20883:α 20873:α 20869:∑ 20848:α 20841:→ 20831:− 20826:β 20819:→ 20802:β 20799:α 20785:∂ 20776:β 20770:α 20766:∑ 20762:− 20752:α 20745:˙ 20733:α 20723:α 20719:∑ 20644:δ 20613:α 20567:→ 20490:α 20483:→ 20473:− 20468:β 20461:→ 20444:β 20441:α 20431:β 20425:α 20421:∑ 20417:− 20402:α 20395:˙ 20390:→ 20366:α 20343:α 20339:∑ 20330:∫ 20294:˙ 20289:→ 20263:→ 20242:∫ 20225:→ 20163:− 20157:˙ 20129:˙ 20119:∂ 20111:∂ 20088:∑ 19982:˙ 19966:∑ 19925:− 19908:˙ 19892:∑ 19873:− 19856:˙ 19840:∑ 19820:− 19786:˙ 19776:∂ 19768:∂ 19745:∑ 19689:− 19672:˙ 19656:∑ 19604:˙ 19581:∂ 19564:∂ 19552:∑ 19548:− 19536:¨ 19517:˙ 19501:∑ 19474:− 19457:˙ 19441:∑ 19352:˙ 19236:− 19219:˙ 19192:∑ 19173:∫ 19136:˙ 19104:∫ 19014:≈ 19006:⋯ 19002:− 18991:φ 18964:φ 18959:ν 18955:∂ 18949:μ 18945:∂ 18938:∂ 18934:∂ 18917:ν 18913:∂ 18903:φ 18892:ν 18888:∂ 18866:φ 18861:ν 18857:∂ 18851:μ 18847:∂ 18840:∂ 18836:∂ 18823:− 18817:φ 18790:φ 18785:μ 18781:∂ 18774:∂ 18770:∂ 18760:− 18755:μ 18740:μ 18736:∂ 18709:ε 18660:φ 18633:φ 18628:μ 18624:∂ 18617:∂ 18613:∂ 18603:− 18598:μ 18583:μ 18579:∂ 18575:ε 18572:∫ 18569:≈ 18537:φ 18508:φ 18503:μ 18499:∂ 18490:∂ 18486:∂ 18476:ε 18471:μ 18467:∂ 18444:ε 18436:∫ 18399:φ 18390:ε 18382:μ 18378:∂ 18356:φ 18351:μ 18347:∂ 18340:∂ 18336:∂ 18320:φ 18311:ε 18293:φ 18290:∂ 18286:∂ 18271:∫ 18229:ε 18220:∫ 18194:ε 18156:μ 18146:μ 18142:∂ 18138:≈ 18099:φ 17981:μ 17968:− 17956:φ 17927:− 17918:φ 17868:φ 17863:μ 17859:∂ 17852:∂ 17848:∂ 17833:μ 17780:μ 17765:μ 17761:∂ 17757:≈ 17715:− 17607:μ 17592:μ 17588:∂ 17584:≈ 17539:μ 17524:μ 17520:∂ 17516:≈ 17457:off shell 17425:⋅ 17422:∂ 17403:∫ 17372:≈ 17367:μ 17354:μ 17341:∂ 17337:∫ 17298:integrate 17268:μ 17260:− 17254:φ 17239:φ 17234:μ 17230:∂ 17223:∂ 17211:∂ 17198:μ 17168:μ 17133:≈ 17123:μ 17115:− 17109:φ 17094:φ 17089:μ 17085:∂ 17078:∂ 17066:∂ 17053:μ 17049:∂ 17009:μ 16993:μ 16980:∂ 16976:∫ 16972:≈ 16960:μ 16943:φ 16928:φ 16923:μ 16919:∂ 16912:∂ 16900:∂ 16889:∂ 16885:∫ 16859:φ 16839:φ 16834:μ 16830:∂ 16823:∂ 16811:∂ 16803:μ 16799:∂ 16795:− 16789:φ 16786:∂ 16774:∂ 16757:∫ 16712:∫ 16684:theorem, 16671:Lie group 16655:, we say 16647:, we say 16645:off shell 16603:φ 16598:μ 16594:∂ 16578:φ 16508:μ 16498:μ 16494:∂ 16490:≈ 16433:μ 16418:… 16412:φ 16409:∂ 16406:∂ 16400:φ 16397:∂ 16382:φ 16374:μ 16361:∂ 16357:∫ 16353:≈ 16316:∫ 16234:φ 16226:and that 16208:≈ 16193:φ 16190:δ 16182:φ 16169:δ 16143:φ 16095:φ 16043:φ 16011:φ 15949:φ 15944:μ 15940:∂ 15924:φ 15905:∫ 15893:φ 15837:φ 15813:φ 15777:φ 15772:μ 15768:∂ 15761:φ 15717:φ 15621:→ 15506:φ 15499:… 15487:φ 15465:spacetime 15414:, in the 15305:Ψ 15292:σ 15278:φ 15272:∂ 15264:∂ 15254:− 15244:σ 15232:− 15223:φ 15200:σ 15186:φ 15180:∂ 15172:∂ 15156:σ 15117:σ 15104:σ 15096:∂ 15092:∂ 15046:φ 15028:ε 15025:− 15016:Ψ 15012:ε 15000:φ 14993:¯ 14990:δ 14955:μ 14942:μ 14934:∂ 14923:φ 14919:∂ 14904:φ 14858:ν 14848:μ 14815:φ 14784:φ 14753:φ 14705:φ 14687:ε 14675:φ 14668:¯ 14665:δ 14650:Ψ 14646:ε 14630:φ 14626:δ 14617:μ 14609:ε 14597:μ 14589:δ 14538:σ 14530:δ 14520:μ 14507:ν 14493:φ 14478:φ 14457:φ 14450:¯ 14447:δ 14436:σ 14422:φ 14416:∂ 14408:∂ 14392:σ 14384:∂ 14380:∂ 14322:σ 14314:δ 14304:μ 14291:ν 14277:φ 14262:φ 14241:φ 14234:¯ 14231:δ 14220:σ 14206:φ 14200:∂ 14192:∂ 14176:σ 14168:∂ 14164:∂ 14157:Ω 14153:∫ 14108:φ 14101:¯ 14098:δ 14087:σ 14073:φ 14067:∂ 14059:∂ 14043:σ 14035:∂ 14031:∂ 14021:σ 14007:φ 13998:¯ 13995:δ 13984:σ 13970:φ 13964:∂ 13956:∂ 13941:φ 13934:¯ 13931:δ 13916:σ 13902:φ 13896:∂ 13888:∂ 13873:σ 13865:∂ 13861:∂ 13833:μ 13820:ν 13806:φ 13791:φ 13779:− 13769:μ 13756:ν 13742:α 13727:α 13673:φ 13669:∂ 13661:∂ 13643:σ 13629:φ 13623:∂ 13615:∂ 13600:σ 13592:∂ 13588:∂ 13545:φ 13538:¯ 13535:δ 13519:σ 13511:∂ 13507:∂ 13494:σ 13486:∂ 13475:φ 13471:∂ 13462:¯ 13459:δ 13448:σ 13434:φ 13425:¯ 13422:δ 13384:σ 13370:φ 13361:¯ 13358:δ 13347:σ 13333:φ 13327:∂ 13319:∂ 13304:φ 13297:¯ 13294:δ 13279:φ 13275:∂ 13267:∂ 13246:μ 13233:ν 13219:φ 13204:φ 13192:− 13182:μ 13169:ν 13155:α 13140:α 13065:σ 13057:δ 13047:μ 13034:ν 13020:φ 13005:φ 12983:σ 12975:∂ 12971:∂ 12951:μ 12938:ν 12924:φ 12909:φ 12897:− 12887:μ 12874:ν 12860:α 12845:α 12821:Ω 12817:∫ 12777:σ 12769:∂ 12758:φ 12754:∂ 12743:σ 12729:φ 12674:μ 12661:ν 12647:φ 12632:φ 12618:Ω 12614:∫ 12610:− 12607:ξ 12587:μ 12583:ξ 12574:ν 12560:α 12545:α 12529:′ 12525:Ω 12520:∫ 12475:μ 12457:φ 12450:¯ 12447:δ 12432:μ 12414:φ 12401:μ 12383:α 12345:φ 12341:δ 12308:μ 12290:φ 12286:δ 12274:μ 12256:φ 12243:μ 12239:ξ 12225:α 12221:→ 12212:φ 12183:μ 12175:δ 12167:μ 12154:μ 12150:ξ 12146:→ 12141:μ 12087:φ 12049:ε 12046:∂ 12041:φ 12038:∂ 12026:˙ 12015:∂ 12007:∂ 12001:− 11987:− 11981:˙ 11964:˙ 11953:∂ 11945:∂ 11900:∂ 11895:φ 11892:∂ 11881:, we get 11859:ε 11856:∂ 11851:φ 11848:∂ 11836:˙ 11825:∂ 11817:∂ 11811:− 11797:− 11791:˙ 11772:∂ 11767:φ 11764:∂ 11752:˙ 11741:∂ 11733:∂ 11673:ε 11670:∂ 11665:φ 11662:∂ 11650:˙ 11639:∂ 11631:∂ 11625:− 11603:ε 11600:∂ 11595:φ 11592:∂ 11580:˙ 11569:∂ 11561:∂ 11521:˙ 11502:∂ 11497:φ 11494:∂ 11482:˙ 11471:∂ 11463:∂ 11432:˙ 11413:∂ 11408:φ 11405:∂ 11393:˙ 11382:∂ 11374:∂ 11368:− 11327:˙ 11286:− 11245:˙ 11161:˙ 11142:∂ 11139:ε 11136:∂ 11131:φ 11122:∂ 11109:˙ 11098:∂ 11090:∂ 11078:ε 11075:∂ 11070:φ 11067:∂ 11053:∂ 11045:∂ 11033:˙ 11014:∂ 11011:ε 11008:∂ 11003:φ 10994:∂ 10981:˙ 10970:∂ 10962:∂ 10950:ε 10947:∂ 10942:φ 10939:∂ 10922:˙ 10911:∂ 10903:∂ 10866:ε 10863:∂ 10858:φ 10855:∂ 10843:˙ 10832:∂ 10824:∂ 10758:˙ 10739:∂ 10736:ε 10733:∂ 10728:φ 10719:∂ 10706:˙ 10695:∂ 10687:∂ 10675:ε 10672:∂ 10667:φ 10664:∂ 10650:∂ 10642:∂ 10611:∫ 10573:˙ 10554:∂ 10549:φ 10546:∂ 10534:˙ 10523:∂ 10515:∂ 10484:˙ 10465:∂ 10460:φ 10457:∂ 10445:˙ 10434:∂ 10426:∂ 10420:− 10379:˙ 10338:− 10297:˙ 10237:ε 10168:¨ 10149:∂ 10144:φ 10141:∂ 10129:˙ 10118:∂ 10110:∂ 10095:˙ 10076:˙ 10047:∂ 10039:φ 10030:∂ 10012:˙ 10001:∂ 9993:∂ 9978:˙ 9959:∂ 9954:φ 9951:∂ 9937:∂ 9929:∂ 9906:¨ 9887:∂ 9882:φ 9879:∂ 9867:˙ 9856:∂ 9848:∂ 9833:˙ 9809:∂ 9804:φ 9801:∂ 9769:˙ 9758:∂ 9750:∂ 9735:˙ 9716:∂ 9711:φ 9708:∂ 9691:˙ 9680:∂ 9672:∂ 9628:˙ 9609:∂ 9604:φ 9601:∂ 9589:˙ 9578:∂ 9570:∂ 9492:¨ 9473:∂ 9468:φ 9465:∂ 9459:− 9453:˙ 9434:∂ 9431:ε 9428:∂ 9423:φ 9414:∂ 9392:˙ 9362:∂ 9354:φ 9345:∂ 9338:− 9324:˙ 9313:∂ 9305:∂ 9288:ε 9285:∂ 9280:φ 9277:∂ 9262:˙ 9243:∂ 9238:φ 9235:∂ 9229:− 9213:∂ 9205:∂ 9174:∫ 9120:˙ 9079:− 9038:˙ 8978:ε 8938:, we get 8875:ε 8872:− 8855:˙ 8841:ε 8829:ε 8826:− 8796:∂ 8791:φ 8788:∂ 8776:ε 8764:ε 8761:− 8739:φ 8725:ε 8704:ε 8687:∫ 8626:˙ 8576:ε 8555:ε 8538:∫ 8524:ε 8474:ε 8471:− 8454:˙ 8440:ε 8428:ε 8425:− 8395:∂ 8390:φ 8387:∂ 8375:ε 8352:φ 8310:˙ 8298:→ 8283:˙ 8234:ε 8222:ε 8219:− 8197:φ 8188:ε 8165:φ 8135:→ 8107:ε 8087:→ 8022:∂ 8014:∂ 7990:˙ 7979:∂ 7971:∂ 7897:˙ 7838:∫ 7773:μ 7770:ν 7766:η 7757:ψ 7749:μ 7741:∂ 7734:∗ 7730:ψ 7726:∂ 7720:− 7715:∗ 7711:ψ 7702:μ 7694:∂ 7689:ψ 7686:∂ 7667:ν 7627:∗ 7623:ψ 7619:ψ 7601:μ 7598:ν 7594:η 7588:∗ 7584:ψ 7578:μ 7574:∂ 7570:ψ 7565:ν 7561:∂ 7480:∗ 7476:ψ 7470:θ 7464:− 7456:→ 7451:∗ 7447:ψ 7434:ψ 7429:θ 7418:→ 7415:ψ 7302:σ 7282:μ 7251:ν 7239:ν 7230:μ 7190:ν 7185:μ 7181:δ 7177:− 7172:μ 7165:φ 7161:⋅ 7149:ν 7142:φ 7138:∂ 7126:∂ 7108:ν 7101:φ 7097:∂ 7085:∂ 7079:φ 7074:σ 7070:∂ 7064:σ 7059:μ 7055:δ 7039:ν 7034:μ 7030:δ 7026:− 7018:ν 7009:μ 6961:μ 6921:μ 6877:μ 6868:δ 6864:− 6856:μ 6847:Λ 6816:μ 6807:δ 6797:ε 6793:− 6788:μ 6768:↦ 6759:μ 6717:φ 6712:μ 6708:∂ 6702:μ 6693:δ 6689:− 6677:Ψ 6651:μ 6624:μ 6616:ε 6613:− 6608:μ 6581:μ 6549:μ 6540:δ 6530:ε 6526:− 6521:μ 6508:φ 6505:↦ 6497:μ 6486:φ 6462:δ 6440:μ 6431:δ 6421:ε 6412:μ 6404:↦ 6399:μ 6363:μ 6351:φ 6345:μ 6341:∂ 6333:φ 6283:ν 6273:ν 6269:∂ 6230:index or 6218:ν 6181:Ψ 6177:⋅ 6169:ν 6162:φ 6158:∂ 6146:∂ 6140:− 6135:ν 6126:Λ 6117:ν 6053:μ 6044:Λ 6038:μ 6034:∂ 6024:ε 6010:↦ 5968:Ψ 5958:ε 5951:φ 5948:↦ 5945:φ 5913:… 5863:μ 5859:Λ 5853:μ 5849:∂ 5845:ε 5832:↦ 5754:Ψ 5734:φ 5712:μ 5687:Ψ 5661:Ψ 5658:ε 5652:φ 5649:↦ 5646:φ 5623:φ 5568:μ 5557:φ 5552:μ 5548:∂ 5541:φ 5526:∫ 5452:φ 5436:mechanics 5378:⋅ 5357:× 5344:⋅ 5323:× 5310:⋅ 5294:⋅ 5285:˙ 5274:∂ 5266:∂ 5229:× 5181:= 0, and 5154:× 5145:θ 5142:δ 5131:→ 5030:˙ 5014:∂ 5006:∂ 4889:− 4883:˙ 4872:⋅ 4863:˙ 4852:∂ 4844:∂ 4746:⋅ 4737:˙ 4726:∂ 4718:∂ 4712:− 4691:− 4685:˙ 4674:⋅ 4665:˙ 4654:∂ 4646:∂ 4581:generator 4565:generator 4504:ε 4494:∑ 4478:δ 4455:ε 4445:∑ 4431:δ 4356:δ 4340:′ 4330:→ 4311:δ 4297:′ 4289:→ 4183:ignorable 4151:˙ 4141:∂ 4133:∂ 4033:˙ 4023:∂ 4015:∂ 3946:∂ 3938:∂ 3919:˙ 3908:∂ 3900:∂ 3799:˙ 3732:˙ 3707:∫ 3615:collision 3555:˙ 3501:invariant 3452:∂ 3402:Δ 3379:∂ 3323:Δ 3297:∂ 3090:ψ 3024:˙ 3000:˙ 2990:∂ 2979:∂ 2965:∑ 2961:− 2911:˙ 2889:˙ 2879:∂ 2871:∂ 2865:− 2851:± 2848:≈ 2831:˙ 2821:Δ 2806:˙ 2796:∂ 2788:∂ 2782:∫ 2765:± 2762:≈ 2756:Δ 2732:Δ 2703:↦ 2671:φ 2653:˙ 2643:∂ 2632:∂ 2618:∑ 2591:φ 2574:↦ 2495:˙ 2486:∂ 2475:∂ 2430:φ 2419:˙ 2410:∂ 2399:∂ 2350:φ 2341:˙ 2332:∂ 2324:∂ 2289:φ 2280:˙ 2271:∂ 2263:∂ 2232:Δ 2153:φ 2144:˙ 2135:∂ 2127:∂ 2121:± 2112:δ 2103:˙ 2094:∂ 2086:∂ 2080:± 2074:≈ 2064:τ 2057:δ 2051:± 2037:˙ 2028:∂ 2020:∂ 2014:∫ 2011:≈ 2005:˙ 1996:Δ 1987:˙ 1978:∂ 1970:∂ 1964:∫ 1961:≈ 1955:Δ 1952:∫ 1943:Δ 1920:˙ 1911:Δ 1898:˙ 1889:∂ 1878:∂ 1868:≈ 1862:Δ 1839:τ 1828:δ 1825:± 1819:˙ 1810:→ 1804:˙ 1775:δ 1772:≫ 1769:τ 1758:δ 1738:τ 1715:δ 1695:τ 1684:δ 1661:˙ 1632:˙ 1583:∫ 1537:φ 1514:Δ 1506:bring no 1425:τ 1387:φ 1377:variation 1297:δ 1288:↦ 1279:φ 1168:Lie group 1080:invariant 896:Malliavin 783:Geometric 682:Laplacian 632:Dirichlet 543:Geometric 128:− 75:∫ 23841:Symmetry 23566:(2002). 23457:(2016). 23427:(1993). 23376:(2010). 23268:10019667 23168:(1980). 23135:(1970). 22924:37903527 22755:See also 22694:rotation 19033:Examples 18701:for all 17436:⟩ 17399:⟨ 17322:on shell 17316:Comments 17294:symmetry 16686:on shell 16668:symmetry 16653:on shell 16641:on shell 16534:for all 16248:on shell 16061:subspace 16025:boundary 15734:integral 15474:-valued 14572:symmetry 10228:′ 8969:′ 8906:′ 8891:′ 8868:′ 8822:′ 8757:′ 8672:′ 8657:′ 8643:′ 8632:′ 8607:′ 8596:′ 8517:′ 8467:′ 8421:′ 8327:′ 8316:′ 8215:′ 8155:′ 8144:′ 8094:′ 7541:spinless 6250:index). 5064:globally 4970:momentum 4781:Examples 4260:and the 4201:symmetry 3599:momentum 3232:-domain, 1152:symmetry 1046:and the 936:Glossary 846:Advanced 824:Jacobian 778:Exterior 708:Gradient 700:Theorems 667:Gradient 606:Integral 568:Binomial 553:Harmonic 418:improper 414:Integral 371:Integral 353:Reynolds 328:Quotient 257:Concepts 93:′ 60:Calculus 23777:Bibcode 23696:Bibcode 23629:Bibcode 23596:Bibcode 23516:Bibcode 23312:Bibcode 23276:2588285 23248:Bibcode 22984:Bibcode 22745:entropy 22732:of the 22726:complex 22724:of the 16033:compact 16023:at the 15732:is the 15463:is the 15375:be the 14802:in the 14771:is the 13407:commute 5189:as the 5073:locally 5068:locally 5056:special 4621:Noether 3641:of the 3193:of the 1082:), its 1003:of the 931:History 829:Hessian 718:Stokes' 713:Green's 545: ( 472: ( 416: ( 338:Inverse 313:Product 224: ( 23729: 23534: 23469: 23443: 23411:: 1–12 23388: 23340: 23330: 23274: 23266: 23205: 23178: 23143: 23110: 23075: 23027: 23002: 22955: 22945: 22922: 22912: 22887: 22877: 21940:. And 21404:where 21078:Then, 19719:Then, 19066:, is: 19060:action 19058:. The 18173:Then, 17815:. So, 17794:where 17623:Then, 17419: 15825:, its 15697:action 15646:action 15351:. Let 14730:where 12506:action 8256:where 7778: 7531:, the 7485: 7443: 7437: 7396:field 7357:, the 6454:(with 5679:where 5443:fields 4820:energy 4606:length 4547:where 4525: 4364: 4088: 3957: 3830: 3754: 3687: 3677:action 3591:per se 3567: 3485:, and 2077: 1350:action 1105:energy 1005:action 773:Tensor 768:Matrix 655:Vector 573:Taylor 531:Series 168:Limits 23767:arXiv 23742:arXiv 23707:(PDF) 23686:arXiv 23674:(PDF) 23650:arXiv 23550:arXiv 23532:S2CID 23506:arXiv 23338:S2CID 23272:S2CID 23238:arXiv 22730:gauge 22342:Then 15736:over 15450:space 15395:over 15383:from 7523:and − 7214:with 6835:, so 6208:field 4614:angle 4577:, and 4401:such 3507:(its 1415:] 1401:[ 1226:with 1196:up to 1174:of a 1027:of a 596:Ratio 563:Power 482:Euler 460:Discs 455:Parts 323:Power 318:Chain 247:total 23727:ISBN 23579:2020 23467:ISBN 23441:ISBN 23417:2020 23386:ISBN 23366:2020 23328:ISBN 23290:link 23264:PMID 23203:ISBN 23176:ISBN 23141:ISBN 23108:ISBN 23073:ISBN 23025:ISBN 23000:ISSN 22953:OCLC 22943:ISBN 22920:OCLC 22910:ISBN 22885:OCLC 22875:ISBN 22851:1918 22736:and 22683:time 21819:For 20773:< 20673:And 20553:For 20530:for 20428:< 17553:and 17478:and 16643:and 15472:real 14775:of 8059:flow 7539:for 5726:and 5058:and 4556:are 4390:and 3609:and 3601:and 3513:time 3262:and 2197:and 1103:and 677:Curl 637:Abel 601:Root 23785:doi 23637:doi 23604:doi 23524:doi 23320:doi 23256:doi 23100:doi 23065:doi 22992:doi 22747:of 22708:In 18034:not 18010:any 16261:on 16135:at 16111:of 16063:of 16055:as 16031:is 16027:if 15849:in 15557:to 15455:In 15448:of 15410:In 15399:.) 15387:to 15379:of 15075:at 7384:= | 7338:.) 7294:as 5612:). 5054:In 4798:+ δ 4777:). 4573:of 3394:to 1851:. 1099:of 308:Sum 23802:: 23783:. 23775:. 23761:. 23694:. 23682:52 23680:. 23676:. 23635:. 23625:72 23623:. 23619:. 23602:. 23592:76 23590:. 23570:. 23530:. 23522:. 23514:. 23500:. 23465:. 23461:. 23439:. 23431:. 23407:. 23384:. 23357:. 23353:. 23336:. 23326:. 23318:. 23286:}} 23282:{{ 23270:. 23262:. 23254:. 23246:. 23234:52 23232:. 23155:^ 23122:^ 23106:. 23071:. 22998:. 22990:. 22980:01 22978:. 22974:. 22951:. 22918:. 22883:. 22859:^ 22845:. 22751:. 22740:. 22657:. 21607:. 21575:0. 19312:dt 19306:= 19062:, 19017:0. 18721:. 18040:= 18018:QS 17976:12 17828:12 17799:12 17775:12 17463:. 17375:0. 17136:0. 16673:. 16292:, 16254:) 15478:, 15459:, 15343:, 14878:, 8061:, 7525:iψ 7521:iψ 7509:δθ 5187:δθ 5105:δθ 5098:× 5094:= 4944:δq 4942:+ 4933:→ 4794:→ 4616:. 4413:. 4388:δt 4219:. 3867:, 3780:, 3653:. 3491:A 3481:, 2747:: 1408:, 1198:a 1182:. 1039:. 480:, 476:, 23791:. 23787:: 23779:: 23769:: 23763:6 23750:. 23744:: 23735:. 23716:. 23698:: 23688:: 23658:. 23652:: 23643:. 23639:: 23631:: 23610:. 23606:: 23598:: 23581:. 23558:. 23552:: 23538:. 23526:: 23518:: 23508:: 23502:1 23475:. 23449:. 23419:. 23399:. 23394:. 23368:. 23344:. 23322:: 23314:: 23292:) 23278:. 23258:: 23250:: 23240:: 23211:. 23184:. 23149:. 23116:. 23102:: 23081:. 23067:: 23033:. 23006:. 22994:: 22986:: 22959:. 22926:. 22891:. 22634:0 22631:= 22622:j 22584:. 22580:) 22574:4 22532:2 22529:1 22523:( 22513:x 22505:) 22498:+ 22476:x 22471:( 22454:= 22438:f 22431:] 22425:[ 22422:Q 22418:] 22412:L 22404:) 22388:( 22375:[ 22371:= 22358:j 22327:. 22322:L 22311:x 22307:= 22298:f 22270:) 22264:L 22253:x 22248:( 22234:= 22230:] 22224:4 22210:x 22168:x 22162:2 22159:1 22153:[ 22116:. 22112:) 22105:+ 22083:x 22078:( 22072:3 22061:4 22054:) 22037:+ 22005:x 22001:+ 21984:( 21967:= 21964:] 21959:L 21954:[ 21951:Q 21905:. 21902:) 21899:x 21896:( 21890:+ 21887:) 21884:x 21881:( 21859:x 21855:= 21852:] 21849:) 21846:x 21843:( 21837:[ 21834:Q 21821:Q 21800:x 21795:4 21791:d 21786:) 21780:4 21738:2 21735:1 21729:( 21722:= 21712:x 21707:4 21703:d 21698:] 21694:) 21691:x 21688:( 21672:, 21669:) 21666:x 21663:( 21656:[ 21650:L 21642:= 21635:] 21629:[ 21624:S 21572:= 21567:M 21564:C 21549:x 21540:M 21528:P 21519:0 21516:= 21510:t 21507:d 21496:j 21490:d 21462:M 21459:C 21449:x 21435:M 21415:P 21383:M 21380:C 21370:x 21363:M 21357:t 21348:P 21342:= 21331:) 21315:x 21302:m 21295:t 21275:x 21260:m 21255:( 21241:= 21225:f 21215:] 21200:x 21193:[ 21183:Q 21173:) 21167:L 21142:x 21123:( 21109:= 21096:j 21063:. 21048:x 21035:m 21021:= 21012:f 20984:i 20975:x 20965:m 20948:t 20945:d 20941:d 20912:. 20907:i 20892:x 20879:m 20865:= 20854:) 20838:x 20816:x 20808:( 20795:V 20789:i 20781:t 20757:i 20742:x 20729:m 20715:= 20708:] 20703:L 20698:[ 20693:i 20689:Q 20658:. 20653:j 20648:i 20640:t 20637:= 20633:] 20629:) 20626:t 20623:( 20618:j 20609:x 20604:[ 20598:i 20594:Q 20564:Q 20538:N 20511:, 20508:t 20505:d 20501:] 20496:) 20480:x 20458:x 20450:( 20437:V 20412:2 20407:) 20387:x 20378:( 20371:2 20362:m 20354:N 20349:1 20346:= 20334:[ 20327:= 20317:t 20314:d 20310:] 20306:) 20303:t 20300:( 20286:x 20278:, 20275:) 20272:t 20269:( 20260:x 20253:[ 20247:L 20239:= 20231:] 20222:x 20216:[ 20210:S 20166:L 20152:i 20148:x 20136:i 20126:x 20114:L 20103:3 20098:1 20095:= 20092:i 20061:0 20058:= 20055:t 20052:d 20048:/ 20044:j 20041:d 20014:. 20011:) 20008:x 20005:( 20002:V 19999:+ 19994:2 19989:i 19979:x 19970:i 19960:2 19957:m 19952:= 19941:] 19937:) 19934:x 19931:( 19928:V 19920:2 19915:i 19905:x 19896:i 19886:2 19883:m 19877:[ 19868:2 19863:i 19853:x 19844:i 19836:m 19833:= 19823:L 19817:] 19812:i 19808:x 19804:[ 19801:Q 19793:i 19783:x 19771:L 19760:3 19755:1 19752:= 19749:i 19741:= 19734:j 19704:. 19701:) 19698:x 19695:( 19692:V 19684:2 19679:i 19669:x 19660:i 19650:2 19647:m 19642:= 19639:L 19611:i 19601:x 19589:i 19585:x 19576:) 19573:x 19570:( 19567:V 19556:i 19543:i 19533:x 19524:i 19514:x 19505:i 19497:m 19494:= 19490:] 19486:) 19483:x 19480:( 19477:V 19469:2 19464:i 19454:x 19445:i 19435:2 19432:m 19426:[ 19419:t 19416:d 19412:d 19407:= 19404:] 19401:L 19398:[ 19395:Q 19382:V 19378:x 19364:) 19361:t 19358:( 19349:x 19343:= 19340:] 19337:) 19334:t 19331:( 19328:x 19325:[ 19322:Q 19310:/ 19308:d 19304:Q 19272:. 19269:t 19266:d 19261:) 19257:) 19254:) 19251:t 19248:( 19245:x 19242:( 19239:V 19231:2 19226:i 19216:x 19207:3 19202:1 19199:= 19196:i 19186:2 19183:m 19177:( 19170:= 19160:t 19157:d 19152:] 19148:) 19145:t 19142:( 19133:x 19127:, 19124:) 19121:t 19118:( 19115:x 19111:[ 19107:L 19101:= 19094:] 19091:x 19088:[ 19083:S 19064:S 19056:t 19052:V 19048:x 19044:m 19010:] 18998:] 18994:] 18988:[ 18985:Q 18981:] 18975:L 18967:) 18941:( 18928:[ 18923:[ 18909:+ 18906:] 18900:[ 18897:Q 18883:] 18877:L 18869:) 18843:( 18830:[ 18826:2 18820:] 18814:[ 18811:Q 18807:] 18801:L 18793:) 18777:( 18764:[ 18751:f 18746:[ 18682:x 18677:n 18673:d 18667:} 18663:] 18657:[ 18654:Q 18650:] 18644:L 18636:) 18620:( 18607:[ 18594:f 18589:{ 18559:x 18554:n 18550:d 18544:} 18540:] 18534:[ 18531:Q 18527:] 18521:L 18512:) 18494:( 18480:[ 18463:+ 18460:] 18455:L 18450:[ 18447:Q 18440:{ 18433:= 18423:x 18418:n 18414:d 18409:} 18405:) 18402:] 18396:[ 18393:Q 18387:( 18373:] 18367:L 18359:) 18343:( 18330:[ 18326:+ 18323:] 18317:[ 18314:Q 18307:) 18301:L 18280:( 18275:{ 18268:= 18258:x 18253:n 18249:d 18245:] 18240:L 18235:[ 18232:] 18226:[ 18223:q 18217:= 18210:] 18205:S 18200:[ 18197:] 18191:[ 18188:q 18152:f 18135:] 18130:L 18125:[ 18122:Q 18086:x 18078:x 18076:( 18074:q 18070:x 18066:x 18064:( 18062:q 18058:ε 18054:q 18050:Q 18048:) 18046:x 18044:( 18042:ε 18038:q 18026:ε 18022:ε 18014:Q 17986:. 17972:f 17965:) 17962:] 17959:] 17953:[ 17948:1 17944:Q 17940:[ 17935:2 17931:Q 17924:] 17921:] 17915:[ 17910:2 17906:Q 17902:[ 17897:1 17893:Q 17889:( 17885:) 17879:L 17871:) 17855:( 17842:( 17838:= 17824:j 17813:2 17810:Q 17806:1 17803:Q 17796:f 17771:f 17754:] 17751:] 17746:L 17741:[ 17736:1 17732:Q 17728:[ 17723:2 17719:Q 17712:] 17709:] 17704:L 17699:[ 17694:2 17690:Q 17686:[ 17681:1 17677:Q 17673:= 17670:] 17665:L 17660:[ 17657:] 17652:2 17648:Q 17644:, 17639:1 17635:Q 17631:[ 17602:2 17598:f 17581:] 17576:L 17571:[ 17566:2 17562:Q 17534:1 17530:f 17513:] 17508:L 17503:[ 17498:1 17494:Q 17483:2 17480:Q 17476:1 17473:Q 17443:0 17440:= 17430:J 17416:x 17411:4 17407:d 17363:s 17359:d 17350:J 17344:N 17310:M 17273:, 17264:f 17257:] 17251:[ 17248:Q 17242:) 17226:( 17216:L 17204:= 17194:J 17164:J 17129:] 17119:f 17112:] 17106:[ 17103:Q 17097:) 17081:( 17071:L 17059:[ 17035:N 17014:. 17005:s 17000:d 16989:f 16983:N 16956:s 16951:d 16946:] 16940:[ 16937:Q 16931:) 16915:( 16905:L 16892:N 16881:+ 16878:x 16873:n 16868:d 16862:] 16856:[ 16853:Q 16849:] 16842:) 16826:( 16816:L 16779:L 16767:[ 16761:N 16753:= 16745:] 16741:x 16736:n 16731:d 16723:L 16716:N 16707:[ 16703:Q 16678:N 16661:Q 16657:Q 16649:Q 16624:. 16621:] 16618:x 16615:, 16612:) 16609:x 16606:( 16590:, 16587:) 16584:x 16581:( 16575:[ 16570:L 16565:= 16562:) 16559:x 16556:( 16551:L 16536:x 16519:) 16516:x 16513:( 16504:f 16487:] 16484:) 16481:x 16478:( 16473:L 16468:[ 16465:Q 16452:N 16429:s 16425:d 16421:] 16415:, 16403:, 16394:, 16391:) 16388:x 16385:( 16379:[ 16370:f 16364:N 16349:] 16345:x 16340:n 16335:d 16327:L 16320:N 16311:[ 16307:Q 16294:Q 16271:C 16211:0 16202:) 16199:x 16196:( 16185:] 16179:[ 16174:S 16121:S 16073:C 16057:x 16029:M 15984:. 15981:x 15976:n 15972:d 15967:] 15964:x 15961:, 15958:) 15955:x 15952:( 15936:, 15933:) 15930:x 15927:( 15921:[ 15916:L 15909:M 15900:= 15896:] 15890:[ 15885:S 15859:C 15786:) 15783:x 15780:, 15764:, 15758:( 15753:L 15738:M 15720:] 15714:[ 15709:S 15679:C 15657:R 15629:, 15625:R 15616:C 15611:: 15606:S 15586:. 15572:3 15567:R 15539:m 15534:R 15510:m 15502:, 15496:, 15491:1 15469:m 15461:M 15431:R 15420:M 15404:M 15397:M 15389:T 15385:M 15361:C 15349:T 15345:M 15337:n 15315:. 15309:A 15300:) 15289:, 15282:A 15267:L 15258:( 15250:] 15240:X 15235:L 15227:A 15217:X 15211:L 15197:, 15190:A 15175:L 15165:[ 15161:= 15152:j 15125:0 15122:= 15113:j 15100:x 15077:ε 15073:ε 15056:. 15050:A 15040:X 15034:L 15020:A 15009:= 15004:A 14961:. 14951:X 14938:x 14927:A 14913:= 14908:A 14898:X 14892:L 14866:0 14863:= 14855:, 14844:X 14819:A 14804:X 14788:A 14757:A 14747:X 14741:L 14709:A 14699:X 14693:L 14684:+ 14679:A 14659:= 14654:A 14643:= 14634:A 14613:X 14606:= 14593:x 14555:. 14551:0 14548:= 14544:} 14534:x 14526:) 14516:x 14512:, 14504:, 14497:A 14487:, 14482:A 14473:( 14469:L 14466:+ 14461:A 14433:, 14426:A 14411:L 14401:{ 14388:x 14352:. 14348:0 14345:= 14342:x 14337:4 14333:d 14328:} 14318:x 14310:) 14300:x 14296:, 14288:, 14281:A 14271:, 14266:A 14257:( 14253:L 14250:+ 14245:A 14217:, 14210:A 14195:L 14185:{ 14172:x 14122:. 14118:) 14112:A 14084:, 14077:A 14062:L 14052:( 14039:x 14026:= 14018:, 14011:A 13981:, 13974:A 13959:L 13950:+ 13945:A 13924:) 13913:, 13906:A 13891:L 13882:( 13869:x 13852:= 13844:] 13839:) 13829:x 13825:, 13817:, 13810:A 13800:, 13795:A 13786:( 13782:L 13775:) 13765:x 13761:, 13753:, 13746:A 13736:, 13731:A 13722:( 13718:L 13714:[ 13677:A 13664:L 13655:= 13651:) 13640:, 13633:A 13618:L 13609:( 13596:x 13560:. 13555:) 13549:A 13528:( 13515:x 13502:= 13490:x 13479:A 13453:= 13445:, 13438:A 13390:. 13381:, 13374:A 13344:, 13337:A 13322:L 13313:+ 13308:A 13283:A 13270:L 13261:= 13257:] 13252:) 13242:x 13238:, 13230:, 13223:A 13213:, 13208:A 13199:( 13195:L 13188:) 13178:x 13174:, 13166:, 13159:A 13149:, 13144:A 13135:( 13131:L 13127:[ 13100:. 13096:0 13093:= 13090:x 13085:4 13081:d 13076:} 13071:] 13061:x 13053:) 13043:x 13039:, 13031:, 13024:A 13014:, 13009:A 13000:( 12996:L 12992:[ 12979:x 12966:+ 12962:] 12957:) 12947:x 12943:, 12935:, 12928:A 12918:, 12913:A 12904:( 12900:L 12893:) 12883:x 12879:, 12871:, 12864:A 12854:, 12849:A 12840:( 12836:L 12832:[ 12827:{ 12786:. 12773:x 12762:A 12748:= 12740:, 12733:A 12700:0 12697:= 12694:x 12689:4 12685:d 12680:) 12670:x 12666:, 12658:, 12651:A 12641:, 12636:A 12627:( 12623:L 12602:4 12598:d 12593:) 12579:, 12571:, 12564:A 12554:, 12549:A 12540:( 12536:L 12485:. 12480:) 12471:x 12467:( 12461:A 12441:+ 12437:) 12428:x 12424:( 12418:A 12410:= 12406:) 12397:x 12393:( 12387:A 12369:x 12365:α 12349:A 12318:. 12313:) 12304:x 12300:( 12294:A 12283:+ 12279:) 12270:x 12266:( 12260:A 12252:= 12248:) 12235:( 12229:A 12216:A 12179:x 12172:+ 12163:x 12159:= 12137:x 12123:μ 12119:μ 12115:μ 12111:x 12107:A 12091:A 12055:. 12022:q 12010:L 11998:T 11994:) 11990:L 11977:q 11960:q 11948:L 11938:( 11914:1 11911:= 11904:q 11879:q 11832:q 11820:L 11808:T 11804:) 11800:L 11787:q 11776:q 11748:q 11736:L 11726:( 11695:. 11692:] 11687:1 11683:t 11679:[ 11646:q 11634:L 11622:] 11617:2 11613:t 11609:[ 11576:q 11564:L 11555:+ 11543:T 11540:] 11535:1 11531:t 11527:[ 11517:q 11506:q 11478:q 11466:L 11457:+ 11454:T 11451:] 11446:2 11442:t 11438:[ 11428:q 11417:q 11389:q 11377:L 11365:T 11362:] 11357:1 11353:t 11349:, 11346:] 11341:1 11337:t 11333:[ 11323:q 11316:, 11313:] 11308:1 11304:t 11300:[ 11296:q 11292:[ 11289:L 11283:T 11280:] 11275:2 11271:t 11267:, 11264:] 11259:2 11255:t 11251:[ 11241:q 11234:, 11231:] 11226:2 11222:t 11218:[ 11214:q 11210:[ 11207:L 11200:= 11197:0 11167:. 11157:q 11146:q 11126:2 11105:q 11093:L 11084:+ 11057:q 11048:L 11039:= 11029:q 11018:q 10998:2 10977:q 10965:L 10956:+ 10932:) 10918:q 10906:L 10894:t 10891:d 10887:d 10881:( 10877:= 10873:) 10839:q 10827:L 10817:( 10810:t 10807:d 10803:d 10771:. 10768:t 10765:d 10754:q 10743:q 10723:2 10702:q 10690:L 10681:+ 10654:q 10645:L 10632:2 10628:t 10620:1 10616:t 10607:+ 10595:T 10592:] 10587:1 10583:t 10579:[ 10569:q 10558:q 10530:q 10518:L 10509:+ 10506:T 10503:] 10498:2 10494:t 10490:[ 10480:q 10469:q 10441:q 10429:L 10417:T 10414:] 10409:1 10405:t 10401:, 10398:] 10393:1 10389:t 10385:[ 10375:q 10368:, 10365:] 10360:1 10356:t 10352:[ 10348:q 10344:[ 10341:L 10335:T 10332:] 10327:2 10323:t 10319:, 10316:] 10311:2 10307:t 10303:[ 10293:q 10286:, 10283:] 10278:2 10274:t 10270:[ 10266:q 10262:[ 10259:L 10252:= 10249:] 10246:0 10243:[ 10234:d 10225:I 10221:d 10215:= 10212:0 10178:. 10175:T 10164:q 10153:q 10125:q 10113:L 10104:+ 10101:T 10091:q 10083:) 10072:q 10060:2 10056:) 10051:q 10044:( 10034:2 10022:( 10008:q 9996:L 9987:+ 9984:T 9974:q 9963:q 9941:q 9932:L 9923:= 9913:T 9902:q 9891:q 9863:q 9851:L 9842:+ 9839:T 9829:q 9821:) 9813:q 9792:t 9789:d 9785:d 9779:( 9765:q 9753:L 9744:+ 9741:T 9731:q 9720:q 9701:) 9687:q 9675:L 9663:t 9660:d 9656:d 9650:( 9646:= 9638:) 9634:T 9624:q 9613:q 9585:q 9573:L 9563:( 9556:t 9553:d 9549:d 9513:. 9510:t 9507:d 9502:) 9498:T 9488:q 9477:q 9449:q 9438:q 9418:2 9407:+ 9404:T 9399:2 9388:q 9375:2 9371:) 9366:q 9359:( 9349:2 9334:( 9320:q 9308:L 9299:+ 9295:) 9271:+ 9268:T 9258:q 9247:q 9225:( 9217:q 9208:L 9195:2 9191:t 9183:1 9179:t 9170:+ 9158:T 9155:] 9150:1 9146:t 9142:, 9139:] 9134:1 9130:t 9126:[ 9116:q 9109:, 9106:] 9101:1 9097:t 9093:[ 9089:q 9085:[ 9082:L 9076:T 9073:] 9068:2 9064:t 9060:, 9057:] 9052:2 9048:t 9044:[ 9034:q 9027:, 9024:] 9019:2 9015:t 9011:[ 9007:q 9003:[ 9000:L 8993:= 8990:] 8987:0 8984:[ 8975:d 8966:I 8962:d 8956:= 8953:0 8932:ε 8928:ε 8903:t 8899:d 8895:] 8888:t 8884:, 8881:] 8878:T 8865:t 8861:[ 8851:q 8844:] 8838:, 8835:] 8832:T 8819:t 8815:[ 8811:q 8807:[ 8800:q 8782:, 8779:] 8773:, 8770:] 8767:T 8754:t 8750:[ 8746:q 8742:[ 8736:[ 8733:L 8728:T 8722:+ 8717:2 8713:t 8707:T 8701:+ 8696:1 8692:t 8683:= 8669:t 8665:d 8661:] 8654:t 8650:, 8647:] 8640:t 8636:[ 8622:q 8614:, 8611:] 8604:t 8600:[ 8592:q 8587:[ 8584:L 8579:T 8573:+ 8568:2 8564:t 8558:T 8552:+ 8547:1 8543:t 8534:= 8527:] 8521:[ 8514:I 8483:. 8480:] 8477:T 8464:t 8460:[ 8450:q 8443:] 8437:, 8434:] 8431:T 8418:t 8414:[ 8410:q 8406:[ 8399:q 8381:= 8378:] 8372:, 8369:] 8366:t 8363:[ 8359:q 8355:[ 8346:t 8343:d 8339:d 8334:= 8331:] 8324:t 8320:[ 8306:q 8295:] 8292:t 8289:[ 8279:q 8262:T 8258:ε 8237:] 8231:, 8228:] 8225:T 8212:t 8208:[ 8204:q 8200:[ 8194:= 8191:] 8185:, 8182:] 8179:t 8176:[ 8172:q 8168:[ 8162:= 8159:] 8152:t 8148:[ 8140:q 8128:] 8125:t 8122:[ 8118:q 8110:T 8104:+ 8101:t 8098:= 8091:t 8080:t 8063:φ 8042:. 8039:] 8036:t 8033:[ 8026:q 8017:L 8008:= 8005:] 8002:t 7999:[ 7986:q 7974:L 7962:t 7959:d 7955:d 7925:t 7922:d 7918:] 7915:t 7912:, 7909:] 7906:t 7903:[ 7893:q 7886:, 7883:] 7880:t 7877:[ 7873:q 7869:[ 7866:L 7859:2 7855:t 7847:1 7843:t 7834:= 7831:I 7819:q 7781:, 7761:) 7745:x 7698:x 7679:( 7675:i 7672:= 7663:j 7649:j 7632:. 7614:2 7610:m 7606:+ 7557:= 7554:L 7517:Ψ 7513:ε 7505:θ 7488:, 7467:i 7460:e 7440:, 7426:i 7422:e 7402:ψ 7398:ψ 7390:ψ 7386:ψ 7382:p 7374:φ 7370:x 7366:x 7364:( 7362:ψ 7351:φ 7347:Ψ 7322:T 7259:0 7256:= 7248:, 7226:T 7197:L 7169:, 7157:) 7146:, 7131:L 7120:( 7116:= 7105:, 7090:L 7051:+ 7046:L 7023:= 7005:T 6981:T 6941:r 6910:μ 6906:T 6884:L 6872:r 6861:= 6851:r 6822:) 6811:r 6801:r 6784:x 6779:( 6773:L 6764:) 6755:x 6751:( 6745:L 6720:. 6697:r 6686:= 6681:r 6647:x 6620:X 6604:x 6577:x 6555:) 6544:r 6534:r 6517:x 6512:( 6502:) 6493:x 6489:( 6435:r 6425:r 6417:+ 6408:x 6395:x 6384:N 6369:) 6359:x 6355:, 6337:, 6328:( 6324:L 6294:, 6291:0 6288:= 6279:j 6238:r 6185:r 6166:, 6151:L 6130:r 6122:= 6112:r 6108:j 6081:N 6058:. 6048:r 6028:r 6020:+ 6015:L 6005:L 5977:, 5972:r 5962:r 5954:+ 5922:, 5919:N 5916:, 5910:, 5907:2 5904:, 5901:1 5898:= 5895:r 5868:, 5842:+ 5837:L 5827:L 5800:L 5776:S 5708:x 5664:, 5655:+ 5606:n 5589:x 5584:4 5580:d 5574:) 5564:x 5560:, 5544:, 5537:( 5531:L 5523:= 5518:S 5489:) 5486:t 5483:, 5479:x 5475:( 5472:T 5415:L 5411:n 5407:n 5403:L 5386:. 5382:L 5374:n 5370:= 5366:) 5361:p 5353:r 5348:( 5340:n 5336:= 5332:) 5327:r 5319:n 5314:( 5306:p 5302:= 5298:Q 5281:q 5269:L 5237:. 5233:r 5225:n 5221:= 5217:Q 5203:Q 5199:q 5195:r 5191:ε 5183:N 5179:T 5162:. 5158:r 5150:n 5139:+ 5135:r 5127:r 5109:n 5100:p 5096:r 5092:L 5039:. 5025:k 5021:q 5009:L 5000:= 4995:k 4991:p 4977:k 4973:p 4965:k 4961:Q 4957:T 4953:N 4948:k 4939:k 4935:q 4930:k 4926:q 4921:k 4917:q 4895:. 4892:L 4879:q 4859:q 4847:L 4838:= 4835:H 4823:H 4816:Q 4812:T 4808:N 4804:q 4800:t 4796:t 4792:t 4756:r 4751:Q 4733:q 4721:L 4707:r 4703:T 4698:) 4694:L 4681:q 4661:q 4649:L 4639:( 4625:N 4610:q 4601:r 4597:Q 4588:r 4584:Q 4570:r 4568:T 4553:r 4549:ε 4528:, 4520:r 4515:Q 4508:r 4498:r 4490:= 4482:q 4469:r 4465:T 4459:r 4449:r 4441:= 4434:t 4411:N 4407:r 4399:N 4395:q 4392:δ 4367:, 4360:q 4353:+ 4349:q 4345:= 4335:q 4322:q 4314:t 4308:+ 4305:t 4302:= 4293:t 4282:t 4265:q 4258:t 4254:L 4196:k 4194:q 4189:k 4187:q 4158:k 4148:q 4136:L 4127:= 4122:k 4118:p 4091:, 4085:0 4082:= 4076:t 4073:d 4066:k 4062:p 4058:d 4052:= 4048:) 4040:k 4030:q 4018:L 4009:( 4002:t 3999:d 3995:d 3979:k 3977:q 3960:. 3950:q 3941:L 3932:= 3928:) 3915:q 3903:L 3894:( 3887:t 3884:d 3880:d 3861:I 3857:t 3855:( 3853:q 3833:. 3824:t 3821:d 3815:q 3811:d 3805:= 3795:q 3778:q 3774:q 3757:, 3751:t 3748:d 3744:) 3741:t 3738:, 3728:q 3721:, 3717:q 3713:( 3710:L 3704:= 3701:I 3689:L 3681:I 3669:q 3570:. 3564:0 3561:= 3552:X 3546:= 3540:t 3537:d 3532:X 3529:d 3505:X 3497:X 3455:U 3430:r 3426:j 3405:S 3382:U 3354:r 3350:j 3326:S 3300:U 3275:1 3271:t 3248:0 3244:t 3220:) 3217:t 3214:, 3209:r 3205:q 3201:( 3181:U 3161:] 3156:1 3152:t 3148:, 3143:0 3139:t 3135:[ 3112:) 3109:t 3106:, 3101:r 3097:q 3093:( 3070:) 3067:t 3064:( 3061:q 3037:) 3031:r 3021:q 3013:) 3007:r 2997:q 2986:/ 2982:L 2975:( 2969:r 2958:L 2954:( 2950:T 2928:, 2924:) 2918:r 2908:q 2896:r 2886:q 2874:L 2862:L 2858:( 2854:T 2844:) 2838:r 2828:q 2813:r 2803:q 2791:L 2779:+ 2776:L 2773:T 2769:( 2759:S 2735:S 2712:T 2709:+ 2706:t 2700:t 2689:. 2675:r 2666:) 2660:r 2650:q 2639:/ 2635:L 2628:( 2622:r 2595:r 2587:+ 2582:r 2578:q 2569:r 2565:q 2542:r 2538:q 2509:p 2506:= 2502:) 2492:q 2482:/ 2478:L 2471:( 2450:q 2426:) 2416:q 2406:/ 2402:L 2395:( 2374:, 2371:) 2366:1 2362:t 2358:( 2354:) 2338:q 2327:L 2317:( 2313:= 2310:) 2305:0 2301:t 2297:( 2293:) 2277:q 2266:L 2256:( 2235:S 2210:1 2206:t 2183:0 2179:t 2156:. 2141:q 2130:L 2118:= 2115:q 2100:q 2089:L 2070:) 2060:q 2047:( 2034:q 2023:L 2002:q 1984:q 1973:L 1958:L 1949:= 1946:S 1917:q 1906:) 1895:q 1885:/ 1881:L 1873:( 1865:L 1835:/ 1831:q 1816:q 1801:q 1778:q 1765:/ 1761:q 1718:q 1691:/ 1687:q 1658:q 1629:q 1606:q 1586:L 1580:= 1577:S 1557:L 1517:S 1494:] 1489:1 1485:t 1481:, 1476:0 1472:t 1468:[ 1448:S 1413:1 1410:t 1406:0 1403:t 1359:S 1332:) 1329:t 1326:( 1323:q 1300:q 1294:+ 1291:q 1285:q 1282:: 1259:q 1232:ρ 1228:ρ 1223:ρ 1219:G 1211:I 1117:X 1113:X 985:e 978:t 971:v 549:) 484:) 420:) 228:) 140:) 137:a 134:( 131:f 125:) 122:b 119:( 116:f 113:= 110:t 107:d 103:) 100:t 97:( 90:f 84:b 79:a 34:. 20:)

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Index