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153:, defined by the physics of the problem. These approaches answer questions relating starting and ending points: Which trajectory will place a basketball in the hoop? If we launch a rocket to the Moon today, how can it land there in 5 days? The Newtonian and action-principle forms are equivalent, and either one can solve the same problems, but selecting the appropriate form will make solutions much easier.
164:, the difference between kinetic and potential energy. The kinetic energy combines the energy of motion for all the objects in the system; the potential energy depends upon the instantaneous position of the objects and drives the motion of the objects. The motion of the objects places them in new positions with new potential energy values, giving a new value for the Lagrangian.
171:, with 3 space and 3 momentum coordinates for each object in the scenario; energy is a scalar magnitude combining information from all objects, giving an immediate simplification in many cases. The components of force vary with coordinate systems; the energy value is the same in all coordinate systems. Force requires an inertial frame of reference; once velocities approach the
1666:
636:
Action principles apply to a wide variety of physical problems, including all of fundamental physics. The only major exceptions are cases involving friction or when only the initial position and velocities are given. Different action principles have different meaning for the variations; each specific
65:
The action principles have applications as broad as physics, including many problems in classical mechanics but especially in modern problems of quantum mechanics and general relativity. These applications built up over two centuries as the power of the method and its further mathematical development
2727:
For every path, the action integral builds in value from zero at the starting point to its final value at the end. Any nearby path has similar values at similar distances from the starting point. Lines or surfaces of constant partial action value can be drawn across the paths, creating a wave-like
599:
states that any conserved quantities in a
Lagrangian imply a continuous symmetry and conversely. For examples, a Lagrangian independent of time corresponds to a system with conserved energy; spatial translation independence implies momentum conservation; angular rotation invariance implies angular
407:
A system moving between two points takes one particular path; other similar paths are not taken. Each path corresponds to a value of the action. An action principle predicts or explains that the particular path taken has a stationary value for the system's action: similar paths near the one taken
49:
Several distinct action principles differ in the constraints on their initial and final conditions. The names of action principles have evolved over time and differ in details of the endpoints of the paths and the nature of the variation. Quantum action principles generalize and justify the older
212:
399:) energy expressions depend upon the physics problem, and their value at each point on the path depends upon relative coordinates corresponding to that point. The energy function is called a Lagrangian; in simple problems it is the kinetic energy minus the potential energy of the system.
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is fixed during the variation, but not the time, the reverse of the constraints on
Hamilton's principle. Consequently, the same path and end points take different times and energies in the two forms. The solutions in the case of this form of Maupertuis's principle are
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subtract. Close to the path expected from classical physics, phases tend to align; the tendency is stronger for more massive objects that have larger values of action. In the classical limit, one path dominates – the path of stationary action.
2204:
1915:
For quantum mechanics, the action principles have significant advantages: only one mechanical postulate is needed, if a covariant
Lagrangian is used in the action, the result is relativistically correct, and they transition clearly to classical equivalents.
667:
applies. For example, to score points in basketball the ball must leave the shooters hand and go through the hoop, but the time of the flight is not constrained. Maupertuis's least action principle is written mathematically as the stationary condition
637:
application of an action principle requires a specific
Lagrangian describing the physics. A common name for any or all of these principles is "the principle of least action". For a discussion of the names and historical origin of these principles see
2629:
45:
to the action. The action depends on the energy function, and the energy function depends on the position, motion, and interactions in the system: variation of the action allows the derivation of the equations of motion without vector or forces.
191:, and show vectors of forces and velocities. The explanatory diagrams of action-based mechanics have two points with actual and possible paths connecting them. These diagrammatic conventions reiterate the different strong points of each method.
812:
1004:
534:, but not a maximum. Elliptical planetary orbits provide a simple example of two paths with equal action – one in each direction around the orbit; neither can be the minimum or "least action". The path variation implied by
1904:
The concepts and many of the methods useful for particle mechanics also apply to continuous fields. The action integral runs over a
Lagrangian density, but the concepts are so close that the density is often simply called the Lagrangian.
350:
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Action principles assign a number—the action—to each possible path between two points. This number is computed by adding an energy value for each small section of the path multiplied by the time spent in that section:
89:. The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
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applies. For example, imagine planning a trip to the Moon. During your voyage the Moon will continue its orbit around the Earth: it's a moving target. Hamilton's principle for objects at positions
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GarcĂa-Morales, Vladimir; Pellicer, Julio; Manzanares, JosĂ© A. (2008). "Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators".
1661:{\displaystyle (\delta {\mathcal {S}})_{\Delta t}=0,\ \mathrm {where} \ {\mathcal {S}}\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt.}
141:
This approach to mechanics focuses on a single point in space and time, attempting to answer the question: "What happens next?". Mechanics based on action principles begin with the concept of
33:
lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a
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232:. The mathematical technology and terminology of action principles can be learned by thinking in terms of physical space, then applied in the more powerful and general abstract spaces.
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Depending on the action principle, the two points connected by paths in a diagram may represent two particle positions at different times, or the two points may represent values in a
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profoundly affects mechanics based on forces. In action principles, relativity merely requires a different
Lagrangian: the principle itself is independent of coordinate systems.
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All of the paths contribute in the quantum action principle. At the end point, where the paths meet, the paths with similar phases add, and those with phases differing by
1134:
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and which was later fully developed in
Hamilton's ingenious optico-mechanical theory. This analogy played a fundamental role in the development of modern wave-mechanics.
2412:, the phase changes rapidly along the path: the amplitude averages to a small number. Thus the Planck constant sets the boundary between classical and quantum mechanics.
397:
375:
3156:
Thomas A. Moore "Least-Action
Principle" in Macmillan Encyclopedia of Physics, John Rigden, editor, Simon & Schuster Macmillan, 1996, volume 2, page 840.
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When quantum effects are important, new action principles are needed. Instead of a particle following a path, quantum mechanics defines a probability amplitude
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The
Schwinger form makes analysis of variation of the Lagrangian itself, for example, variation in potential source strength, especially transparent.
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577:. The action integral depends on the coordinates of the objects, and these coordinates depend upon the path taken. Thus the action integral is a
659:
Dwyane Wade shooting free throws illustrating the kind of physical constraints suitable for application of
Maupertuis' least action principle
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69:
This article introduces the action principle concepts and summarizes other articles with more details on concepts and specific principles.
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momentum conservation. These examples are global symmetries, where the independence is itself independent of space or time; more general
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is the difference between kinetic energy and potential energy at each point on the path. Solution of the resulting equations gives the
3130:
17:
4199:
Biot, Maurice Anthony (1975). "A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics".
3210:
4298:
4444:
Nakane, Michiyo, and Craig G. Fraser. "The Early History of Hamilton-Jacobi Dynamics 1834–1837." Centaurus 44.3-4 (2002): 161–227.
3341:
2801:), a way of decomposing the computed electron density of molecules in to atoms as a way of gaining insight into chemical bonding.
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1931:. Feynman's integral method was not a variational principle but reduces to the classical least action principle; it led to his
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describing the physical system. The accumulated value of this energy function between two states of the system is called the
167:
Using energy rather than force gives immediate advantages as a basis for mechanics. Force mechanics involves 3-dimensional
3312:
2817:
applied the principle of least action to derive the field equations of general relativity. His action, now known as the
2742:... thus pointed to that remarkable analogy between optical and mechanical phenomena which was observed much earlier by
3324:
671:
4529:
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1332:. For a rigid body with no net force, the actions are identical, and the variational principles become equivalent to
4367:
the law of reflection is stated, namely that incoming and outgoing rays form the same angle with the surface normal.
2199:{\displaystyle \psi (x_{k+1},t+\varepsilon )={\frac {1}{A}}\int e^{iS(x_{k+1},x_{k})/\hbar }\psi (x_{k},t)\,dx_{k},}
3179:. Dover books on physics and chemistry (Republ. of the 3rd ed., publ. in 1968 ed.). New York, NY: Dover Publ.
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is the classical action. Instead of single path with stationary action, all possible paths add (the integral over
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1935:. Schwinger's differential approach relates infinitesimal amplitude changes to infinitesimal action changes.
225:
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2624:{\displaystyle \delta (q_{r_{\text{f}}}|q_{r_{\text{i}}})=i(q_{r_{\text{f}}}|\delta S|q_{r_{\text{i}}}),}
1863:
1758:
1720:
3575:
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
441:
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1944:
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1423:
807:{\displaystyle W\ {\stackrel {\text{def}}{=}}\ \int _{q_{1}}^{q_{2}}\mathbf {p} \cdot \mathbf {dq} ,}
664:
650:
51:
4454:
Mehra, Jagdish (1987). "Einstein, Hilbert, and the Theory of Gravitation". In Mehra, Jagdish (ed.).
1952:
408:
have very similar action value. This variation in the action value is key to the action principles.
2381:
2291:
617:
578:
2785:, e.g. the shape of elastic rods under load, the shape of a liquid between two vertical plates (a
1717:
means that we only consider paths taking the same time, as well as connecting the same two points
1109:
999:{\displaystyle p_{k}\ {\stackrel {\text{def}}{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}},}
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2740:
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1417:
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358:
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1165:: functions relating coordinates to each other in which time is simply an index or a parameter.
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The Development of the Action Principle: A Didactic History from Euler-Lagrange to Schwinger
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Surfaces of constant action shown as wavefronts perpendicular to paths for the case of light
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187:
The explanatory diagrams in force-based mechanics usually focus on a single point, like the
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Introductory study of mechanics, the science of interacting objects, typically begins with
59:
34:
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8:
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2782:
2719:
1136:, to emphasize that the variation used in this form of the action principle differs from
345:{\displaystyle S=\int _{t_{1}}^{t_{2}}{\big (}{\text{KE}}(t)-{\text{PE}}(t){\big )}\,dt,}
86:
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4135:
4124:"From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective"
4123:
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55:
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demonstrated how this principle can be used in quantum calculations by discerning the
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When the physics problem gives the two endpoints as a position and a time, that is as
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150:
142:
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38:
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Quantum action principles are used in the quantum theory of atoms in molecules (
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3027:
2890:{\displaystyle S={\frac {1}{2\kappa }}\int R{\sqrt {-g}}\,\mathrm {d} ^{4}x,}
2814:
1408:
A path to the Moon must account for the motion of the Moon during the voyage.
4212:
3499:"Hamilton's Principle and the Conservation Theorems of Mathematical Physics"
530:
is zero. For action principles, the stationary point may be a minimum or a
4358:
4185:
4034:
Rojo, Alberto; Bloch, Anthony, eds. (2018). "Relativity and Least Action".
1404:
605:
531:
3795:
4524:
J. S. Schwinger, Quantum Kinematics and Dynamics, W. A. Benjamin (1970),
4083:
3259:
The lazy universe : an introduction to the principle of least action
3003:
229:
4350:
3745:
2769:
Action principles are applied to derive differential equations like the
4479:
3122:
2038:
related to a probability amplitude at a different point later in time:
1928:
1829:
4507:
R. Feynman, Quantum Mechanics and Path Integrals, McGraw-Hill (1965),
4158:
3966:. SpringerBriefs in Physics. Cham: Springer International Publishing.
3475:
3365:
3234:
2728:
view of the action. Analysis like this connects particle-like rays of
199:
156:
The energy function in the action principles is not the total energy (
2786:
2450:
Schwinger's approach relates variations in the transition amplitudes
3571:"Which Symmetry? Noether, Weyl, and Conservation of Electric Charge"
3072:
later showed that this path has the shortest length and least time.
4140:
2326:. The phase of the amplitude is given by the action divided by the
604:
symmetries having a functional dependence on space or time lead to
3099:
developed a variational form for classical mechanics known as the
3546:
The road to reality: a complete guide to the laws of the universe
3141:
independently applied this principle in quantum electrodynamics.
609:
4068:
3868:"A 21st century perspective as a primer to introductory physics"
211:
3053:
3045:
1162:
3425:
Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2008).
3209:
Hanc, Jozef; Taylor, Edwin F.; Tuleja, Slavomir (2005-07-01).
2789:), or the motion of a pendulum when its support is in motion.
2781:
Action principles can be directly applied to many problems in
3429:(3rd ed., ed.). San Francisco, Munich: Addison Wesley.
2798:
1862:. Starting with Hamilton's principle, the local differential
621:
102:
4040:. Cambridge: Cambridge University Press. pp. 162–188.
3821:"Space-Time Approach to Non-Relativistic Quantum Mechanics"
2695:{\displaystyle S=\int _{t_{\text{i}}}^{t_{\text{f}}}L\,dt.}
1927:
developed quantum action principles based on early work by
106:
3729:"Progress in classical and quantum variational principles"
3060:
that, for the path of light reflecting from a mirror, the
50:
classical principles. Action principles are the basis for
4329:
Helzberger, Max (1966). "Optics from Euclid to Huygens".
3342:"Getting the most action out of least action: A proposal"
1168:
3170:
3168:
3166:
3164:
3162:
907:{\displaystyle \mathbf {p} =(p_{1},p_{2},\ldots ,p_{N})}
644:
105:, defined by the acceleration it causes when applied to
3637:(7. print ed.). Cambridge: Cambridge Univ. Press.
3319:(2nd ed.). Cambridge: Cambridge University Press.
3262:. Oxford; New York, NY : Oxford University Press.
1866:
can be derived for systems of fixed energy. The action
1053:{\displaystyle L(\mathbf {q} ,{\dot {\mathbf {q} }},t)}
3997:"The Quantum Mechanical Basis of Conceptual Chemistry"
3612:(Impression: 3 ed.). Oxford: Oxford Univ. Press.
2900:
contained a relativistically invariant volume element
3177:
Variational principles in dynamics and quantum theory
3159:
3044:
The action principle is preceded by earlier ideas in
2972:
2952:
2906:
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2421:
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1997:
1955:
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1342:
1289:
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1112:
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924:
914:
are the particle momenta or the conjugate momenta of
847:
820:
719:
674:
560:
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496:
444:
417:
383:
361:
251:
115:
4447:
3727:
Gray, C. G.; Karl, G.; Novikov, V. A. (2004-02-01).
2358:. When the action of a particle is much larger than
4121:
3424:
4413:
4380:
4277:The Principle of Least Action in Quantum Mechanics
4037:The Principle of Least Action: History and Physics
3915:"Quantum physics explains Newton's laws of motion"
3204:
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4416:Mathematical Thought from Ancient to Modern Times
4383:Mathematical Thought from Ancient to Modern Times
4122:Gay-Balmaz, François; Yoshimura, Hiroaki (2018).
3211:"Variational mechanics in one and two dimensions"
435:so an action principle appears mathematically as
4539:
3310:
3208:
3175:Yourgrau, Wolfgang; Mandelstam, Stanley (1979).
2773:or as direct applications to physical problems.
2708:
3726:
3701:
3193:
2936:{\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x}
2439:
2288:), weighted by a complex probability amplitude
663:When total energy and the endpoints are fixed,
4420:. New York: Oxford University Press. pp.
4387:. New York: Oxford University Press. pp.
3311:Kleppner, Daniel; Kolenkow, Robert J. (2014).
3249:
1938:
327:
289:
4438:
3913:Ogborn, Jon; Taylor, Edwin F. (2004-12-24).
3912:
3420:
3418:
3416:
3414:
3412:
3110:applied the variational principle to derive
3040:History of variational principles in physics
1908:
3955:
3953:
3951:
3702:Kibble, T. W. B.; Berkshire, F. H. (2004).
3387:
3385:
3383:
3313:"Chapter 3: Forces and equations of motion"
3255:
2498:to variations in an action matrix element:
2491:{\displaystyle (q_{\text{f}}|q_{\text{i}})}
1256:{\displaystyle \Delta S=\Delta W-E\Delta t}
595:An important result from geometry known as
4328:
3656:
3654:
3610:The quantum story: a history in 40 moments
3450:Gray, C. G.; Taylor, Edwin F. (May 2007).
3449:
584:
4491:Physikalische Zeitschrift der Sowjetunion
4405:
4259:
4175:
4157:
4139:
4082:
4033:
4001:Monatshefte für Chemie – Chemical Monthly
3744:
3686:
3632:
3601:
3409:
3392:"Ch. 19: The Principle of Least Action".
3281:
3279:
2917:
2868:
2765:Lagrangian (field theory) § Examples
2682:
2179:
1893:
1648:
1199:relates simply to the abbreviated action
332:
3959:
3948:
3706:(5th ed.). Imperial College Press.
3633:Hand, Louis N.; Finch, Janet D. (2008).
3380:
2718:
1890:of the action in Maupertuis' principle.
1403:
654:
620:in 1953 to construct a gauge theory for
4458:(Reprint ed.). Dordrecht: Reidel.
4274:
3818:
3787:An Introduction To Quantum Field Theory
3651:
3607:
3568:
3562:
3543:
3304:
3288:The variational principles of mechanics
3285:
3006:that it is widely applied including in
1384:{\displaystyle \delta (t_{2}-t_{1})=0.}
554:is not the same as a differential like
14:
4540:
3865:
3859:
3783:
3276:
2998:The action principle is so central in
2776:
1179:For time-invariant system, the action
1169:Time-independent potentials; no forces
631:
235:
52:Feynman's version of quantum mechanics
4484:"The Lagrangian in Quantum Mechanics"
4478:
4453:
4411:
4378:
3994:
3339:
2993:
2804:
645:Fixed endpoints with conserved energy
217:Diagrammatic aid for action principle
4456:The physicist's conception of nature
4233:
4198:
3660:
3496:
2715:Hamilton's optico-mechanical analogy
2446:Schwinger's quantum action principle
1710:{\displaystyle \Delta t=t_{2}-t_{1}}
1325:{\displaystyle \Delta t=t_{2}-t_{1}}
182:
92:
4299:"Principle of Least Action – damtp"
1786:{\displaystyle \mathbf {q} (t_{2})}
1748:{\displaystyle \mathbf {q} (t_{1})}
665:Maupertuis's least action principle
24:
4275:Feynman, Richard Phillips (1942),
4201:Bulletin de la Classe des sciences
3995:Bader, Richard F. W. (June 2005).
2920:
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1113:
968:
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72:
25:
4564:
3866:Curtis, Lorenzo J. (2011-09-01).
3784:Peskin, Michael E. (2018-01-31).
2393:
2345:
2311:
2149:
476:{\displaystyle (\delta A)_{C}=0,}
402:
27:Fundamental mechanical principles
2254:{\displaystyle S(x_{k+1},x_{k})}
1839:
1763:
1725:
1620:
1600:
1527:
1428:
1099:{\displaystyle (\delta W)_{E}=0}
1031:
1020:
849:
797:
794:
786:
727:
703:{\displaystyle (\delta W)_{E}=0}
624:, leading some decades later to
210:
198:
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3537:
3490:
3443:
3395:The Feynman Lectures on Physics
3340:Moore, Thomas A. (2004-04-01).
3290:(4 ed.). New York: Dover.
3127:quantum mechanical underpinning
3075:Building on the early work of
2988:Einstein gravitational constant
2809:Inspired by Einstein's work on
2758:
1886:in Hamilton's principle is the
1855:{\displaystyle \mathbf {q} (t)}
1444:{\displaystyle \mathbf {q} (t)}
1393:
608:. The observed conservation of
355:where the form of the kinetic (
158:conserved in an isolated system
3733:Reports on Progress in Physics
3333:
3256:Coopersmith, Jennifer (2017).
3150:
2615:
2594:
2583:
2562:
2553:
2532:
2511:
2485:
2471:
2457:
2248:
2216:
2176:
2157:
2141:
2109:
2079:
2048:
1984:{\displaystyle \psi (x_{k},t)}
1978:
1959:
1849:
1843:
1780:
1767:
1742:
1729:
1645:
1636:
1630:
1610:
1604:
1596:
1531:
1523:
1472:
1458:
1438:
1432:
1372:
1346:
1081:
1071:
1047:
1016:
901:
856:
731:
723:
685:
675:
626:modern particle physics theory
490:, the variation of the action
455:
445:
322:
316:
305:
299:
81:" approaches rather than the "
41:. Action principles apply the
13:
1:
3819:Feynman, R. P. (1948-04-01).
3661:Gray, Chris G. (2009-12-09).
3595:10.1016/S1355-2198(01)00033-8
3548:. New York: Alfred A. Knopf.
3144:
2813:, the renowned mathematician
2709:The optico-mechanical analogy
2634:where the action operator is
2405:{\displaystyle S/\hbar \gg 1}
2319:{\displaystyle e^{iS/\hbar }}
1451:is written mathematically as
431:is used to indicate the path
145:, an energy tradeoff between
3317:An Introduction to Mechanics
3133:of amplitudes. Subsequently
2792:
2440:Schwinger's action principle
1129:{\displaystyle \Delta S_{0}}
581:, a function of a function.
510:with some fixed constraints
7:
4236:"Principle of least action"
3872:European Journal of Physics
3663:"Principle of least action"
3569:Brading, Katherine (1941).
3456:American Journal of Physics
3346:American Journal of Physics
3286:Lanczos, Cornelius (1986).
3215:American Journal of Physics
1418:Hamilton's action principle
918:, defined by the equation
392:{\displaystyle {\text{PE}}}
370:{\displaystyle {\text{KE}}}
205:Diagrammatic aid for forces
10:
4569:
3934:10.1088/0031-9120/40/1/001
3892:10.1088/0143-0807/32/5/014
3763:10.1088/0034-4885/67/2/R02
3497:Hill, E. L. (1951-07-01).
3452:"When action is not least"
3037:
3033:
2762:
2712:
2443:
1942:
1939:Feynman's action principle
1897:
1397:
1219:on the stationary path as
1172:
648:
588:
4261:10.4249/scholarpedia.8291
4101:10.1016/j.aop.2008.04.007
4046:10.1017/9781139021029.007
4013:10.1007/s00706-005-0307-x
3972:10.1007/978-3-030-69105-9
3960:Dittrich, Walter (2021).
3845:10.1103/RevModPhys.20.367
3825:Reviews of Modern Physics
3790:. Boca Raton: CRC Press.
3688:10.4249/scholarpedia.8291
3523:10.1103/RevModPhys.23.253
3503:Reviews of Modern Physics
3089:principle of least action
2734:Huygens–Fresnel principle
1945:Path integral formulation
1909:Quantum action principles
1900:Lagrangian (field theory)
18:Principle of least action
3129:of the principle in the
3101:Hamilton–Jacobi equation
2771:Euler–Lagrange equations
2351:{\displaystyle S/\hbar }
1140:. Here the total energy
101:based on the concept of
4213:10.3406/barb.1975.57878
3608:Baggott, J. E. (2013).
3544:Penrose, Roger (2005).
3121:In 1933, the physicist
3077:Pierre Louis Maupertuis
2979:{\displaystyle \kappa }
2819:Einstein–Hilbert action
2732:with the wavefronts of
1888:Legendre transformation
1864:Euler–Lagrange equation
1064:. Some textbooks write
916:generalized coordinates
585:Conservation principles
547:{\displaystyle \delta }
424:{\displaystyle \delta }
77:Action principles are "
4412:Kline, Morris (1972).
4379:Kline, Morris (1972).
3093:William Rowan Hamilton
2980:
2960:
2937:
2891:
2756:
2724:
2696:
2625:
2492:
2429:
2406:
2372:
2371:{\displaystyle \hbar }
2352:
2330:or quantum of action:
2320:
2282:
2255:
2200:
2032:
2012:
1985:
1894:Classical field theory
1880:
1856:
1822:
1787:
1749:
1711:
1662:
1445:
1409:
1385:
1326:
1277:
1257:
1213:
1193:
1154:
1130:
1100:
1054:
1000:
908:
835:
808:
704:
660:
651:Maupertuis's principle
639:action principle names
571:
548:
524:
504:
477:
425:
393:
371:
346:
135:
3796:10.1201/9780429503559
3087:defining versions of
3085:Joseph Louis Lagrange
2981:
2961:
2938:
2892:
2738:
2722:
2697:
2626:
2493:
2430:
2407:
2373:
2353:
2321:
2283:
2281:{\displaystyle x_{k}}
2256:
2201:
2033:
2013:
2011:{\displaystyle x_{k}}
1986:
1881:
1857:
1823:
1821:{\displaystyle L=T-V}
1788:
1750:
1712:
1663:
1446:
1407:
1386:
1327:
1278:
1258:
1214:
1194:
1155:
1131:
1101:
1055:
1001:
909:
836:
834:{\displaystyle S_{0}}
809:
705:
658:
572:
549:
525:
505:
478:
426:
394:
372:
347:
136:
134:{\displaystyle F=ma.}
43:calculus of variation
4548:Dynamics (mechanics)
4234:Gray, Chris (2009).
3635:Analytical mechanics
3131:quantum interference
3016:theory of relativity
2970:
2950:
2904:
2828:
2641:
2505:
2454:
2428:{\displaystyle \pi }
2419:
2382:
2362:
2334:
2292:
2265:
2210:
2042:
2022:
1995:
1953:
1870:
1835:
1800:
1759:
1721:
1672:
1455:
1424:
1400:Hamilton's principle
1340:
1287:
1283:and time difference
1267:
1223:
1203:
1183:
1144:
1138:Hamilton's variation
1110:
1068:
1010:
922:
845:
818:
717:
672:
558:
538:
514:
494:
486:meaning that at the
442:
415:
381:
359:
249:
113:
60:quantum field theory
4553:Classical mechanics
4351:10.1364/AO.5.001383
4343:1966ApOpt...5.1383H
4285:1942PhDT.........5F
4252:2009SchpJ...4.8291G
4150:2018Entrp..21....8G
4093:2008AnPhy.323.1844G
3884:2011EJPh...32.1259C
3837:1948RvMP...20..367F
3755:2004RPPh...67..159G
3704:Classical Mechanics
3679:2009SchpJ...4.8291G
3587:2002SHPMP..33....3B
3515:1951RvMP...23..253H
3468:2007AmJPh..75..434G
3427:Classical mechanics
3358:2004AmJPh..72..522M
3227:2005AmJPh..73..603H
3066:angle of reflection
2966:. The scale factor
2783:classical mechanics
2777:Classical mechanics
2678:
1592:
814:(sometimes written
784:
632:Distinct principles
286:
236:Action along a path
226:configuration space
87:Newtonian mechanics
3116:general relativity
3097:Carl Gustav Jacobi
3070:Hero of Alexandria
3062:angle of incidence
2994:Other applications
2976:
2956:
2933:
2887:
2811:general relativity
2805:General relativity
2730:geometrical optics
2725:
2692:
2650:
2621:
2488:
2425:
2402:
2368:
2348:
2316:
2278:
2251:
2196:
2028:
2008:
1981:
1876:
1852:
1818:
1783:
1745:
1707:
1658:
1564:
1441:
1410:
1381:
1334:Fermat's principle
1322:
1273:
1253:
1209:
1189:
1175:Fermat's principle
1150:
1126:
1096:
1050:
996:
904:
831:
804:
756:
712:abbreviated action
700:
661:
570:{\displaystyle dt}
567:
544:
520:
500:
473:
421:
389:
367:
342:
258:
189:center of momentum
177:special relativity
131:
56:general relativity
4480:Dirac, Paul A. M.
4465:978-90-277-2536-3
4159:10.3390/e21010008
4071:Annals of Physics
4055:978-0-521-86902-7
3981:978-3-030-69104-2
3922:Physics Education
3805:978-0-429-50355-9
3644:978-0-521-57572-0
3619:978-0-19-956684-6
3555:978-0-679-45443-4
3476:10.1119/1.2710480
3436:978-0-201-65702-9
3366:10.1119/1.1646133
3297:978-0-486-65067-8
3269:978-0-19-874304-0
3235:10.1119/1.1848516
3186:978-0-486-63773-0
3020:quantum mechanics
2959:{\displaystyle R}
2915:
2866:
2850:
2674:
2662:
2610:
2577:
2548:
2526:
2482:
2467:
2093:
2031:{\displaystyle t}
1879:{\displaystyle S}
1627:
1563:
1558:
1536:
1515:
1495:
1276:{\displaystyle E}
1212:{\displaystyle W}
1192:{\displaystyle S}
1153:{\displaystyle E}
1038:
991:
981:
956:
951:
949:
937:
755:
750:
748:
736:
597:Noether's theorem
591:Noether's theorem
523:{\displaystyle C}
503:{\displaystyle A}
387:
377:) and potential (
365:
314:
297:
183:Paths, not points
93:Energy, not force
31:Action principles
16:(Redirected from
4560:
4533:
4522:
4516:
4505:
4499:
4498:
4488:
4476:
4470:
4469:
4451:
4445:
4442:
4436:
4435:
4419:
4409:
4403:
4402:
4386:
4376:
4370:
4369:
4337:(9): 1383–1893.
4326:
4320:
4319:
4317:
4316:
4310:
4304:. Archived from
4303:
4295:
4289:
4287:
4272:
4266:
4265:
4263:
4231:
4225:
4224:
4196:
4190:
4189:
4179:
4161:
4143:
4119:
4113:
4112:
4086:
4084:cond-mat/0602186
4077:(8): 1844–1858.
4066:
4060:
4059:
4031:
4025:
4024:
3992:
3986:
3985:
3957:
3946:
3945:
3919:
3910:
3904:
3903:
3878:(5): 1259–1274.
3863:
3857:
3856:
3816:
3810:
3809:
3781:
3775:
3774:
3748:
3724:
3718:
3717:
3699:
3693:
3692:
3690:
3658:
3649:
3648:
3630:
3624:
3623:
3605:
3599:
3598:
3566:
3560:
3559:
3541:
3535:
3534:
3494:
3488:
3487:
3447:
3441:
3440:
3422:
3407:
3406:
3404:
3403:
3389:
3378:
3377:
3337:
3331:
3330:
3308:
3302:
3301:
3283:
3274:
3273:
3253:
3247:
3246:
3206:
3191:
3190:
3172:
3157:
3154:
3135:Julian Schwinger
3114:'s equations of
3024:particle physics
2985:
2983:
2982:
2977:
2965:
2963:
2962:
2957:
2945:scalar curvature
2942:
2940:
2939:
2934:
2929:
2928:
2923:
2916:
2908:
2896:
2894:
2893:
2888:
2880:
2879:
2874:
2867:
2859:
2851:
2849:
2838:
2754:
2701:
2699:
2698:
2693:
2677:
2676:
2675:
2672:
2665:
2664:
2663:
2660:
2630:
2628:
2627:
2622:
2614:
2613:
2612:
2611:
2608:
2597:
2586:
2581:
2580:
2579:
2578:
2575:
2552:
2551:
2550:
2549:
2546:
2535:
2530:
2529:
2528:
2527:
2524:
2497:
2495:
2494:
2489:
2484:
2483:
2480:
2474:
2469:
2468:
2465:
2434:
2432:
2431:
2426:
2411:
2409:
2408:
2403:
2392:
2377:
2375:
2374:
2369:
2357:
2355:
2354:
2349:
2344:
2325:
2323:
2322:
2317:
2315:
2314:
2310:
2287:
2285:
2284:
2279:
2277:
2276:
2260:
2258:
2257:
2252:
2247:
2246:
2234:
2233:
2205:
2203:
2202:
2197:
2192:
2191:
2169:
2168:
2153:
2152:
2148:
2140:
2139:
2127:
2126:
2094:
2086:
2066:
2065:
2037:
2035:
2034:
2029:
2017:
2015:
2014:
2009:
2007:
2006:
1990:
1988:
1987:
1982:
1971:
1970:
1933:Feynman diagrams
1925:Julian Schwinger
1885:
1883:
1882:
1877:
1861:
1859:
1858:
1853:
1842:
1827:
1825:
1824:
1819:
1792:
1790:
1789:
1784:
1779:
1778:
1766:
1754:
1752:
1751:
1746:
1741:
1740:
1728:
1716:
1714:
1713:
1708:
1706:
1705:
1693:
1692:
1667:
1665:
1664:
1659:
1629:
1628:
1623:
1618:
1603:
1591:
1590:
1589:
1579:
1578:
1577:
1561:
1560:
1559:
1557:
1556:
1544:
1539:
1534:
1530:
1522:
1521:
1513:
1512:
1493:
1483:
1482:
1470:
1469:
1450:
1448:
1447:
1442:
1431:
1390:
1388:
1387:
1382:
1371:
1370:
1358:
1357:
1331:
1329:
1328:
1323:
1321:
1320:
1308:
1307:
1282:
1280:
1279:
1274:
1262:
1260:
1259:
1254:
1218:
1216:
1215:
1210:
1198:
1196:
1195:
1190:
1159:
1157:
1156:
1151:
1135:
1133:
1132:
1127:
1125:
1124:
1105:
1103:
1102:
1097:
1089:
1088:
1059:
1057:
1056:
1051:
1040:
1039:
1034:
1029:
1023:
1005:
1003:
1002:
997:
992:
990:
989:
988:
983:
982:
974:
966:
958:
954:
953:
952:
950:
947:
945:
940:
935:
934:
933:
913:
911:
910:
905:
900:
899:
881:
880:
868:
867:
852:
840:
838:
837:
832:
830:
829:
813:
811:
810:
805:
800:
789:
783:
782:
781:
771:
770:
769:
753:
752:
751:
749:
746:
744:
739:
734:
730:
709:
707:
706:
701:
693:
692:
576:
574:
573:
568:
553:
551:
550:
545:
529:
527:
526:
521:
509:
507:
506:
501:
488:stationary point
482:
480:
479:
474:
463:
462:
430:
428:
427:
422:
398:
396:
395:
390:
388:
385:
376:
374:
373:
368:
366:
363:
351:
349:
348:
343:
331:
330:
315:
312:
298:
295:
293:
292:
285:
284:
283:
273:
272:
271:
214:
202:
151:potential energy
140:
138:
137:
132:
21:
4568:
4567:
4563:
4562:
4561:
4559:
4558:
4557:
4538:
4537:
4536:
4523:
4519:
4506:
4502:
4486:
4477:
4473:
4466:
4452:
4448:
4443:
4439:
4432:
4410:
4406:
4399:
4377:
4373:
4327:
4323:
4314:
4312:
4308:
4301:
4297:
4296:
4292:
4273:
4269:
4232:
4228:
4197:
4193:
4120:
4116:
4067:
4063:
4056:
4032:
4028:
3993:
3989:
3982:
3958:
3949:
3917:
3911:
3907:
3864:
3860:
3817:
3813:
3806:
3782:
3778:
3746:physics/0312071
3725:
3721:
3714:
3700:
3696:
3659:
3652:
3645:
3631:
3627:
3620:
3606:
3602:
3567:
3563:
3556:
3542:
3538:
3495:
3491:
3448:
3444:
3437:
3423:
3410:
3401:
3399:
3391:
3390:
3381:
3338:
3334:
3327:
3309:
3305:
3298:
3284:
3277:
3270:
3254:
3250:
3207:
3194:
3187:
3173:
3160:
3155:
3151:
3147:
3139:Richard Feynman
3112:Albert Einstein
3042:
3036:
3012:fluid mechanics
2996:
2971:
2968:
2967:
2951:
2948:
2947:
2924:
2919:
2918:
2907:
2905:
2902:
2901:
2875:
2870:
2869:
2858:
2842:
2837:
2829:
2826:
2825:
2807:
2795:
2779:
2767:
2761:
2755:
2750:
2717:
2711:
2671:
2667:
2666:
2659:
2655:
2654:
2642:
2639:
2638:
2607:
2603:
2602:
2598:
2593:
2582:
2574:
2570:
2569:
2565:
2545:
2541:
2540:
2536:
2531:
2523:
2519:
2518:
2514:
2506:
2503:
2502:
2479:
2475:
2470:
2464:
2460:
2455:
2452:
2451:
2448:
2442:
2420:
2417:
2416:
2388:
2383:
2380:
2379:
2363:
2360:
2359:
2340:
2335:
2332:
2331:
2328:Planck constant
2306:
2299:
2295:
2293:
2290:
2289:
2272:
2268:
2266:
2263:
2262:
2242:
2238:
2223:
2219:
2211:
2208:
2207:
2187:
2183:
2164:
2160:
2144:
2135:
2131:
2116:
2112:
2102:
2098:
2085:
2055:
2051:
2043:
2040:
2039:
2023:
2020:
2019:
2002:
1998:
1996:
1993:
1992:
1966:
1962:
1954:
1951:
1950:
1947:
1941:
1921:Richard Feynman
1911:
1902:
1896:
1871:
1868:
1867:
1838:
1836:
1833:
1832:
1801:
1798:
1797:
1774:
1770:
1762:
1760:
1757:
1756:
1736:
1732:
1724:
1722:
1719:
1718:
1701:
1697:
1688:
1684:
1673:
1670:
1669:
1668:The constraint
1619:
1617:
1616:
1599:
1585:
1581:
1580:
1573:
1569:
1568:
1546:
1545:
1540:
1538:
1537:
1526:
1517:
1516:
1496:
1475:
1471:
1465:
1464:
1456:
1453:
1452:
1427:
1425:
1422:
1421:
1402:
1396:
1366:
1362:
1353:
1349:
1341:
1338:
1337:
1336:of least time:
1316:
1312:
1303:
1299:
1288:
1285:
1284:
1268:
1265:
1264:
1224:
1221:
1220:
1204:
1201:
1200:
1184:
1181:
1180:
1177:
1171:
1145:
1142:
1141:
1120:
1116:
1111:
1108:
1107:
1084:
1080:
1069:
1066:
1065:
1030:
1028:
1027:
1019:
1011:
1008:
1007:
984:
973:
972:
971:
967:
959:
957:
946:
941:
939:
938:
929:
925:
923:
920:
919:
895:
891:
876:
872:
863:
859:
848:
846:
843:
842:
825:
821:
819:
816:
815:
793:
785:
777:
773:
772:
765:
761:
760:
745:
740:
738:
737:
726:
718:
715:
714:
688:
684:
673:
670:
669:
653:
647:
634:
593:
587:
559:
556:
555:
539:
536:
535:
515:
512:
511:
495:
492:
491:
458:
454:
443:
440:
439:
416:
413:
412:
405:
384:
382:
379:
378:
362:
360:
357:
356:
326:
325:
311:
294:
288:
287:
279:
275:
274:
267:
263:
262:
250:
247:
246:
238:
222:
221:
220:
219:
218:
215:
207:
206:
203:
185:
169:vector calculus
114:
111:
110:
95:
75:
73:Common concepts
28:
23:
22:
15:
12:
11:
5:
4566:
4556:
4555:
4550:
4535:
4534:
4517:
4500:
4471:
4464:
4446:
4437:
4430:
4404:
4397:
4371:
4331:Applied Optics
4321:
4290:
4267:
4226:
4191:
4114:
4061:
4054:
4026:
4007:(6): 819–854.
3987:
3980:
3947:
3905:
3858:
3831:(2): 367–387.
3811:
3804:
3776:
3739:(2): 159–208.
3719:
3712:
3694:
3650:
3643:
3625:
3618:
3600:
3561:
3554:
3536:
3509:(3): 253–260.
3489:
3462:(5): 434–458.
3442:
3435:
3408:
3398:. Vol. II
3379:
3352:(4): 522–527.
3332:
3326:978-0521198110
3325:
3303:
3296:
3275:
3268:
3248:
3221:(7): 603–610.
3192:
3185:
3158:
3148:
3146:
3143:
3095:and in tandem
3081:Leonhard Euler
3050:ancient Greece
3038:Main article:
3035:
3032:
3008:thermodynamics
3000:modern physics
2995:
2992:
2975:
2955:
2943:and the Ricci
2932:
2927:
2922:
2914:
2911:
2898:
2897:
2886:
2883:
2878:
2873:
2865:
2862:
2857:
2854:
2848:
2845:
2841:
2836:
2833:
2806:
2803:
2794:
2791:
2778:
2775:
2760:
2757:
2748:
2744:John Bernoulli
2713:Main article:
2710:
2707:
2703:
2702:
2691:
2688:
2685:
2681:
2670:
2658:
2653:
2649:
2646:
2632:
2631:
2620:
2617:
2606:
2601:
2596:
2592:
2589:
2585:
2573:
2568:
2564:
2561:
2558:
2555:
2544:
2539:
2534:
2522:
2517:
2513:
2510:
2487:
2478:
2473:
2463:
2459:
2444:Main article:
2441:
2438:
2424:
2401:
2398:
2395:
2391:
2387:
2367:
2347:
2343:
2339:
2313:
2309:
2305:
2302:
2298:
2275:
2271:
2250:
2245:
2241:
2237:
2232:
2229:
2226:
2222:
2218:
2215:
2195:
2190:
2186:
2182:
2178:
2175:
2172:
2167:
2163:
2159:
2156:
2151:
2147:
2143:
2138:
2134:
2130:
2125:
2122:
2119:
2115:
2111:
2108:
2105:
2101:
2097:
2092:
2089:
2084:
2081:
2078:
2075:
2072:
2069:
2064:
2061:
2058:
2054:
2050:
2047:
2027:
2005:
2001:
1980:
1977:
1974:
1969:
1965:
1961:
1958:
1943:Main article:
1940:
1937:
1910:
1907:
1898:Main article:
1895:
1892:
1875:
1851:
1848:
1845:
1841:
1817:
1814:
1811:
1808:
1805:
1782:
1777:
1773:
1769:
1765:
1744:
1739:
1735:
1731:
1727:
1704:
1700:
1696:
1691:
1687:
1683:
1680:
1677:
1657:
1654:
1651:
1647:
1644:
1641:
1638:
1635:
1632:
1626:
1622:
1615:
1612:
1609:
1606:
1602:
1598:
1595:
1588:
1584:
1576:
1572:
1567:
1555:
1552:
1549:
1543:
1533:
1529:
1525:
1520:
1511:
1508:
1505:
1502:
1499:
1492:
1489:
1486:
1481:
1478:
1474:
1468:
1463:
1460:
1440:
1437:
1434:
1430:
1398:Main article:
1395:
1392:
1380:
1377:
1374:
1369:
1365:
1361:
1356:
1352:
1348:
1345:
1319:
1315:
1311:
1306:
1302:
1298:
1295:
1292:
1272:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1208:
1188:
1173:Main article:
1170:
1167:
1149:
1123:
1119:
1115:
1095:
1092:
1087:
1083:
1079:
1076:
1073:
1049:
1046:
1043:
1037:
1033:
1026:
1022:
1018:
1015:
995:
987:
980:
977:
970:
965:
962:
944:
932:
928:
903:
898:
894:
890:
887:
884:
879:
875:
871:
866:
862:
858:
855:
851:
828:
824:
803:
799:
796:
792:
788:
780:
776:
768:
764:
759:
743:
733:
729:
725:
722:
699:
696:
691:
687:
683:
680:
677:
649:Main article:
646:
643:
633:
630:
614:Chen Ning Yang
589:Main article:
586:
583:
566:
563:
543:
519:
499:
484:
483:
472:
469:
466:
461:
457:
453:
450:
447:
420:
404:
403:Path variation
401:
353:
352:
341:
338:
335:
329:
324:
321:
318:
310:
307:
304:
301:
291:
282:
278:
270:
266:
261:
257:
254:
237:
234:
216:
209:
208:
204:
197:
196:
195:
194:
193:
184:
181:
173:speed of light
147:kinetic energy
130:
127:
124:
121:
118:
94:
91:
85:" approach of
74:
71:
26:
9:
6:
4:
3:
2:
4565:
4554:
4551:
4549:
4546:
4545:
4543:
4531:
4530:0-7382-0303-3
4527:
4521:
4514:
4513:0-07-020650-3
4510:
4504:
4496:
4492:
4485:
4481:
4475:
4467:
4461:
4457:
4450:
4441:
4433:
4431:0-19-501496-0
4427:
4423:
4418:
4417:
4408:
4400:
4398:0-19-501496-0
4394:
4390:
4385:
4384:
4375:
4368:
4366:
4360:
4356:
4352:
4348:
4344:
4340:
4336:
4332:
4325:
4311:on 2015-10-10
4307:
4300:
4294:
4286:
4282:
4278:
4271:
4262:
4257:
4253:
4249:
4245:
4241:
4237:
4230:
4222:
4218:
4214:
4210:
4206:
4202:
4195:
4187:
4183:
4178:
4173:
4169:
4165:
4160:
4155:
4151:
4147:
4142:
4137:
4133:
4129:
4125:
4118:
4110:
4106:
4102:
4098:
4094:
4090:
4085:
4080:
4076:
4072:
4065:
4057:
4051:
4047:
4043:
4039:
4038:
4030:
4022:
4018:
4014:
4010:
4006:
4002:
3998:
3991:
3983:
3977:
3973:
3969:
3965:
3964:
3956:
3954:
3952:
3943:
3939:
3935:
3931:
3927:
3923:
3916:
3909:
3901:
3897:
3893:
3889:
3885:
3881:
3877:
3873:
3869:
3862:
3854:
3850:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3815:
3807:
3801:
3797:
3793:
3789:
3788:
3780:
3772:
3768:
3764:
3760:
3756:
3752:
3747:
3742:
3738:
3734:
3730:
3723:
3715:
3713:9781860944352
3709:
3705:
3698:
3689:
3684:
3680:
3676:
3672:
3668:
3664:
3657:
3655:
3646:
3640:
3636:
3629:
3621:
3615:
3611:
3604:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3565:
3557:
3551:
3547:
3540:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3493:
3485:
3481:
3477:
3473:
3469:
3465:
3461:
3457:
3453:
3446:
3438:
3432:
3428:
3421:
3419:
3417:
3415:
3413:
3397:
3396:
3388:
3386:
3384:
3375:
3371:
3367:
3363:
3359:
3355:
3351:
3347:
3343:
3336:
3328:
3322:
3318:
3314:
3307:
3299:
3293:
3289:
3282:
3280:
3271:
3265:
3261:
3260:
3252:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3205:
3203:
3201:
3199:
3197:
3188:
3182:
3178:
3171:
3169:
3167:
3165:
3163:
3153:
3149:
3142:
3140:
3136:
3132:
3128:
3124:
3119:
3117:
3113:
3109:
3108:David Hilbert
3104:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3073:
3071:
3067:
3063:
3059:
3056:wrote in his
3055:
3051:
3047:
3041:
3031:
3029:
3028:string theory
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2991:
2989:
2973:
2953:
2946:
2930:
2925:
2912:
2909:
2884:
2881:
2876:
2863:
2860:
2855:
2852:
2846:
2843:
2839:
2834:
2831:
2824:
2823:
2822:
2820:
2816:
2815:David Hilbert
2812:
2802:
2800:
2790:
2788:
2784:
2774:
2772:
2766:
2753:
2747:
2745:
2741:
2737:
2735:
2731:
2721:
2716:
2706:
2689:
2686:
2683:
2679:
2668:
2656:
2651:
2647:
2644:
2637:
2636:
2635:
2618:
2604:
2599:
2590:
2587:
2571:
2566:
2559:
2556:
2542:
2537:
2520:
2515:
2508:
2501:
2500:
2499:
2476:
2461:
2447:
2437:
2422:
2413:
2399:
2396:
2389:
2385:
2365:
2341:
2337:
2329:
2307:
2303:
2300:
2296:
2273:
2269:
2243:
2239:
2235:
2230:
2227:
2224:
2220:
2213:
2193:
2188:
2184:
2180:
2173:
2170:
2165:
2161:
2154:
2145:
2136:
2132:
2128:
2123:
2120:
2117:
2113:
2106:
2103:
2099:
2095:
2090:
2087:
2082:
2076:
2073:
2070:
2067:
2062:
2059:
2056:
2052:
2045:
2025:
2003:
1999:
1991:at one point
1975:
1972:
1967:
1963:
1956:
1946:
1936:
1934:
1930:
1926:
1922:
1917:
1913:
1906:
1901:
1891:
1889:
1873:
1865:
1846:
1831:
1815:
1812:
1809:
1806:
1803:
1796:
1775:
1771:
1737:
1733:
1702:
1698:
1694:
1689:
1685:
1681:
1678:
1655:
1652:
1649:
1642:
1639:
1633:
1624:
1613:
1607:
1593:
1586:
1582:
1574:
1570:
1565:
1541:
1490:
1487:
1484:
1479:
1461:
1435:
1419:
1415:
1406:
1401:
1391:
1378:
1375:
1367:
1363:
1359:
1354:
1350:
1343:
1335:
1317:
1313:
1309:
1304:
1300:
1296:
1293:
1270:
1250:
1244:
1241:
1238:
1232:
1229:
1206:
1186:
1176:
1166:
1164:
1147:
1139:
1121:
1117:
1093:
1090:
1085:
1077:
1074:
1063:
1044:
1041:
1035:
1024:
1013:
993:
985:
978:
975:
963:
942:
930:
926:
917:
896:
892:
888:
885:
882:
877:
873:
869:
864:
860:
853:
826:
822:
801:
790:
778:
774:
766:
762:
757:
741:
720:
713:
697:
694:
689:
681:
678:
666:
657:
652:
642:
640:
629:
627:
623:
619:
615:
611:
607:
603:
598:
592:
582:
580:
564:
561:
541:
533:
517:
497:
489:
470:
467:
464:
459:
451:
448:
438:
437:
436:
434:
418:
409:
400:
339:
336:
333:
319:
308:
302:
280:
276:
268:
264:
259:
255:
252:
244:
243:
242:
233:
231:
227:
213:
201:
192:
190:
180:
178:
174:
170:
165:
163:
159:
154:
152:
148:
144:
128:
125:
122:
119:
116:
108:
104:
100:
99:Newton's laws
90:
88:
84:
80:
70:
67:
63:
61:
57:
53:
47:
44:
40:
36:
32:
19:
4520:
4503:
4494:
4490:
4474:
4455:
4449:
4440:
4415:
4407:
4382:
4374:
4364:
4362:
4334:
4330:
4324:
4313:. Retrieved
4306:the original
4293:
4276:
4270:
4246:(12): 8291.
4243:
4240:Scholarpedia
4239:
4229:
4204:
4200:
4194:
4131:
4127:
4117:
4074:
4070:
4064:
4036:
4029:
4004:
4000:
3990:
3962:
3928:(1): 26–34.
3925:
3921:
3908:
3875:
3871:
3861:
3828:
3824:
3814:
3786:
3779:
3736:
3732:
3722:
3703:
3697:
3673:(12): 8291.
3670:
3667:Scholarpedia
3666:
3634:
3628:
3609:
3603:
3578:
3574:
3564:
3545:
3539:
3506:
3502:
3492:
3459:
3455:
3445:
3426:
3400:. Retrieved
3394:
3349:
3345:
3335:
3316:
3306:
3287:
3258:
3251:
3218:
3214:
3176:
3152:
3120:
3105:
3088:
3074:
3057:
3043:
2997:
2899:
2808:
2796:
2780:
2768:
2759:Applications
2739:
2726:
2704:
2633:
2449:
2414:
1948:
1918:
1914:
1912:
1903:
1411:
1394:Fixed events
1178:
662:
635:
618:Robert Mills
612:was used by
606:gauge theory
601:
594:
532:saddle point
485:
410:
406:
354:
239:
223:
186:
166:
155:
96:
83:differential
76:
68:
64:
48:
30:
29:
4497:(1): 64–72.
4207:(1): 6–30.
3581:(1): 3–22.
3064:equals the
3004:mathematics
1263:for energy
411:The symbol
230:phase space
160:), but the
4542:Categories
4365:Catoptrics
4315:2016-07-18
4279:(thesis),
4141:1904.03738
3402:2023-11-03
3145:References
3123:Paul Dirac
3058:Catoptrica
2763:See also:
2752:C. Lanczos
1929:Paul Dirac
1830:world line
1795:Lagrangian
1062:Lagrangian
579:functional
433:variations
162:Lagrangian
35:Lagrangian
4221:0001-4141
4168:1099-4300
4109:118464686
4021:0026-9247
3942:0031-9120
3900:0143-0807
3853:0034-6861
3771:0034-4885
3531:0034-6861
3484:0002-9505
3374:0002-9505
3243:0002-9505
3106:In 1915,
2974:κ
2910:−
2861:−
2853:∫
2847:κ
2793:Chemistry
2787:capillary
2652:∫
2588:δ
2509:δ
2423:π
2397:≫
2394:ℏ
2366:ℏ
2346:ℏ
2312:ℏ
2155:ψ
2150:ℏ
2096:∫
2077:ε
2046:ψ
2018:and time
1957:ψ
1813:−
1695:−
1676:Δ
1625:˙
1566:∫
1477:Δ
1462:δ
1360:−
1344:δ
1310:−
1291:Δ
1248:Δ
1242:−
1236:Δ
1227:Δ
1114:Δ
1075:δ
1036:˙
979:˙
969:∂
961:∂
886:…
841:), where
791:⋅
758:∫
679:δ
542:δ
449:δ
419:δ
309:−
260:∫
4482:(1933).
4359:20057555
4186:33266724
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2749:—
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