726:
6326:
5842:
4387:
6321:{\displaystyle {\begin{aligned}&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{1}(\mathbf {q} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{2}(\mathbf {q} ,\mathbf {P} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{3}(\mathbf {p} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{4}(\mathbf {p} ,\mathbf {P} ,t)\\\end{aligned}}}
4078:
5399:
739:
4382:{\displaystyle {\begin{aligned}\{A,B\}\equiv \{A,B\}_{\mathbf {q} ,\mathbf {p} }&={\frac {\partial A}{\partial \mathbf {q} }}\cdot {\frac {\partial B}{\partial \mathbf {p} }}-{\frac {\partial A}{\partial \mathbf {p} }}\cdot {\frac {\partial B}{\partial \mathbf {q} }}\\&\equiv \sum _{k}{\frac {\partial A}{\partial q_{k}}}{\frac {\partial B}{\partial p_{k}}}-{\frac {\partial A}{\partial p_{k}}}{\frac {\partial B}{\partial q_{k}}}\,,\end{aligned}}}
1075:
975:
3231:
1016:
5329:
1833:
3003:
8744:
8108:
2251:
867:. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-
8511:
5071:
8228:
2116:
9002:
8884:
7355:
1258:
as ensembles of particles that interact with each other, rather considering each particle as an isolated unit. In the vectorial approach, forces must be determined individually for each particle, whereas in the analytical approach it is enough to know one single function which contains implicitly all
1409:
during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of
Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which
1373:
Development of analytical mechanics has two objectives: (i) increase the range of solvable problems by developing standard techniques with a wide range of applicability, and (ii) understand the mathematical structure of mechanics. In the long run, however, (ii) can help (i) more than a concentration
7483:
do not have to be cyclic, the partition between which coordinates enter the
Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian
1369:
Analytical mechanics aims at even more: not at understanding the mathematical structure of a single mechanical problem, but that of a class of problems so wide that they encompass most of mechanics. It concentrates on systems to which
Lagrangian or Hamiltonian equations of motion are applicable and
6446:
3483:
1854:
done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful – since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for
1365:
of the problem. Moreover, an accurate mental or drawn picture can be made for the motion of two bodies, and it can be as real and accurate as the real bodies moving and interacting. In the three-body problem, parameters can also be assigned specific values; however, the solution at these assigned
5048:
1597:
1239:), Newton's approach is difficult to apply. Using a Newtonian approach is possible, under proper precautions, namely isolating each single particle from the others, and determining all the forces acting on it. Such analysis is cumbersome even in relatively simple systems. Newton thought that
960:
The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics.
7463:
7739:
2678:
7231:
8543:
6930:
1410:
relations can be modeled by equations corresponding to the constraints. In the
Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known as
3988:
7936:
2550:
2131:
1330:, and it is not possible to draw a sharp line between 'simple' and 'not simple' functions. If one speaks merely of 'functions', then every mechanical problem is solved as soon as it has been well stated in differential equations, because given the initial conditions and
3226:{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}=\left({\frac {\partial L}{\partial {\dot {q}}_{1}}},{\frac {\partial L}{\partial {\dot {q}}_{2}}},\cdots {\frac {\partial L}{\partial {\dot {q}}_{N}}}\right)=(p_{1},p_{2}\cdots p_{N})\,,}
3827:
3343:
8378:
8113:
2013:
2941:
5656:
1994:
8892:
8751:
11655:
7844:
929:
Analytical mechanics does not introduce new physics and is not more general than
Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used in
4911:
1475:
coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:
5531:
5324:{\displaystyle T((\lambda {\dot {q}}_{i})^{2},(\lambda {\dot {q}}_{j}\lambda {\dot {q}}_{k}),\mathbf {q} )=\lambda ^{2}T(({\dot {q}}_{i})^{2},{\dot {q}}_{j}{\dot {q}}_{k},\mathbf {q} )\,,\quad L(\mathbf {q} ,\mathbf {\dot {q}} )\,,}
7250:
1366:
values or a collection of such solutions does not reveal the mathematical structure of the problem. As in many other problems, the mathematical structure can be elucidated only by examining the differential equations themselves.
6753:
4887:
852:. A scalar is a quantity, whereas a vector is represented by quantity and direction. The results of these two different approaches are equivalent, but the analytical mechanics approach has many advantages for complex problems.
6345:
3382:
1907:
9558:
9500:
4487:
7599:
7378:
1276:, the principle that this action be stationary under small variation of some other mechanical quantity generates the required set of differential equations. The statement of the principle does not require any special
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3582:
2585:
7141:
3632:
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9639:
9191:
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9328:
6813:
1828:{\displaystyle {\frac {d\mathbf {q} }{dt}}=\left({\frac {dq_{1}}{dt}},{\frac {dq_{2}}{dt}},\dots ,{\frac {dq_{N}}{dt}}\right)\equiv \mathbf {\dot {q}} =({\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{N}).}
3587:
which is often considered one of
Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:
6828:
4778:
2352:
9066:
8326:
4604:
3873:
9894:
80:
9848:
2427:
1584:
9777:
9743:
1912:
9222:
863:
the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as
5847:
4083:
8739:{\displaystyle {\mathcal {H}}(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\mathbf {r} ,t)=\sum _{i=1}^{N}{\dot {\phi }}_{i}(\mathbf {r} ,t)\pi _{i}(\mathbf {r} ,t)-{\mathcal {L}}\,.}
3738:
3242:
11600:
9442:
7052:
Routhian mechanics is a hybrid formulation of
Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates. If the Lagrangian of a system has
8103:{\displaystyle {\mathcal {L}}={\mathcal {L}}(\phi _{1},\phi _{2},\dots ,\nabla \phi _{1},\nabla \phi _{2},\dots ,\partial _{t}\phi _{1},\partial _{t}\phi _{2},\ldots ,\mathbf {r} ,t)\,.}
2246:{\displaystyle {\frac {\partial }{\partial \mathbf {q} }}=\left({\frac {\partial }{\partial q_{1}}},{\frac {\partial }{\partial q_{2}}},\dots ,{\frac {\partial }{\partial q_{N}}}\right)}
1232:
and the motion of planets around the Sun. In this approach, Newton's laws describe the motion by a differential equation and then the problem is reduced to the solving of that equation.
2852:
1224:" understood as a single point to which a mass is attached. Newton's method has been successfully applied to a wide range of physical problems, including the motion of a particle in
5548:
898:
on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as
8506:{\displaystyle \pi _{i}(\mathbf {r} ,t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}_{i}}}\,\quad {\dot {\phi }}_{i}\equiv {\frac {\partial \phi _{i}}{\partial t}}}
6554:
1189:
The goal of mechanical theory is to solve mechanical problems, such as arise in physics and engineering. Starting from a physical system—such as a mechanism or a star system—a
11645:
2816:
104:
9597:
9152:
1030:
8538:
8223:{\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}\right)={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}\,,}
6608:
5733:
5701:
11650:
10891:
4816:
2111:{\displaystyle {\boldsymbol {\mathcal {Q}}}={\frac {d}{dt}}\left({\frac {\partial T}{\partial \mathbf {\dot {q}} }}\right)-{\frac {\partial T}{\partial \mathbf {q} }}\,,}
7750:
5379:
1858:
10252:
Kibble, Tom, and Berkshire, Frank H. "Classical
Mechanics" (5th Edition). Singapore, World Scientific Publishing Company, 2004.
9266:
8997:{\displaystyle {\frac {\delta }{\delta \phi _{i}}}={\frac {\partial }{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial }{\partial (\partial _{\mu }\phi _{i})}}}
3366:
8879:{\displaystyle {\dot {\phi }}_{i}=+{\frac {\delta {\mathcal {H}}}{\delta \pi _{i}}}\,,\quad {\dot {\pi }}_{i}=-{\frac {\delta {\mathcal {H}}}{\delta \phi _{i}}}\,,}
5435:
10065:
8330:
Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as
9805:
9709:
7350:{\displaystyle {\dot {\mathbf {q} }}=+{\frac {\partial R}{\partial \mathbf {p} }}\,,\quad {\dot {\mathbf {p} }}=-{\frac {\partial R}{\partial \mathbf {q} }}\,,}
1295:
It is not altogether clear what is meant by 'solving' a set of differential equations. A problem is regarded as solved when the particles coordinates at time
11269:
6441:{\displaystyle \mathbf {\dot {P}} =-{\frac {\partial K}{\partial \mathbf {Q} }}\,,\quad \mathbf {\dot {Q}} =+{\frac {\partial K}{\partial \mathbf {P} }}\,,}
3478:{\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,}
11638:
11274:
6707:
5043:{\displaystyle {\frac {\partial L}{\partial q_{j}}}=0\,\rightarrow \,{\frac {dp_{j}}{dt}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0}
1266:
Still, deriving the equations of motion of a complicated mechanical system requires a unifying basis from which they follow. This is provided by various
9506:
9448:
1361:
has not. The two-body problem is solved by formulas involving parameters; their values can be changed to study the class of all solutions, that is, the
7458:{\displaystyle {\frac {d}{dt}}{\frac {\partial R}{\partial {\dot {\boldsymbol {\zeta }}}}}={\frac {\partial R}{\partial {\boldsymbol {\zeta }}}}\,.}
1270:: behind each set of equations there is a principle that expresses the meaning of the entire set. Given a fundamental and universal quantity called
10884:
4410:
2301:
10998:
9018:
8278:
7734:{\displaystyle {\mathcal {Q}}_{r}={\frac {\partial S}{\partial \alpha _{r}}}\,,\quad S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}^{2}\,,}
7507:
5745:
in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies the
1193:
is developed in the form of a differential equation. The model can be solved numerically or analytically to determine the motion of the system.
11633:
10769:
6620:
4710:
10539:
2673:{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,}
1516:
770:
7226:{\displaystyle R=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {p} ,{\boldsymbol {\zeta }},{\dot {\boldsymbol {\zeta }}})\,,}
3650:
2732:
11451:
3520:
10877:
10858:
4651:) for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time
3594:
1034:
359:
9645:
9603:
11496:
9158:
6944:
1448:
860:
478:
9334:
9272:
6925:{\displaystyle H=H(\mathbf {q} ,\mathbf {p} ,t)=H\left(\mathbf {q} ,{\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }},t\right)}
6768:
3861:)) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the
1263:. However, the analytical treatment does not require the knowledge of these forces and takes these kinematic conditions for granted.
11003:
8331:
3983:{\displaystyle \{(\mathbf {q} (t),\mathbf {p} (t))\in \mathbb {R} ^{2N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {P}}\,,}
1463:; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of
1139:
451:
4900:
If the
Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are
910:. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the
11412:
2545:{\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {q} ,\mathbf {\dot {q}} ,t)-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)}
1111:
6474:". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.
4522:
10737:
9854:
1259:
the forces acting on and in the system. Such simplification is often done using certain kinematic conditions which are stated
36:
9811:
11013:
10490:
10463:
10440:
10421:
10400:
10361:
10278:
10229:
10194:
6485:
is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformation
1092:
988:
895:
7933:
is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:
1118:
11579:
9749:
9715:
9197:
3822:{\displaystyle {\mathcal {P}}={\mathcal {C}}\times {\mathcal {M}}=\{(\mathbf {q} ,\mathbf {p} )\in \mathbb {R} ^{2N}\}\,,}
3338:{\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)}
11033:
11028:
433:
10216:. Encyclopedia of Physics / Handbuch der Physik. Vol. 2 / 3 / 1. Berlin, Heidelberg: Springer Berlin Heidelberg.
1125:
10532:
10510:
10338:
10319:
10299:
10167:
1176:
1158:
1056:
1002:
763:
9403:
11446:
11130:
3514:). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:
1197:
828:
Analytical mechanics was developed by many scientists and mathematicians during the 18th century and onward, after
7468:
Set up in this way, although the
Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian with
5054:" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.
725:
11193:
10953:
6762:
is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:
1107:
1038:
99:
5342:
5057:
If the Lagrangian is time-independent the Hamiltonian is also time-independent (i.e. both are constant in time).
2936:{\displaystyle \{\mathbf {q} (t)\in \mathbb {R} ^{N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {C}}\,,}
11120:
10685:
7498:
7493:
2688:
1096:
907:
354:
94:
5651:{\displaystyle \delta {\mathcal {S}}=\delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0\,,}
1989:{\displaystyle {\boldsymbol {\mathcal {Q}}}=({\mathcal {Q}}_{1},{\mathcal {Q}}_{2},\dots ,{\mathcal {Q}}_{N})}
11704:
11125:
11018:
6463:
5776:
The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of
5407:
2718:
2705:
This formulation identifies the actual path followed by the motion as a selection of the path over which the
1460:
883:
1243:"action equals reaction" would take care of all complications. This is false even for such simple system as
11679:
11023:
10525:
10380:, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
10109:
8253:
second order partial differential equations in the fields, which in general will be coupled and nonlinear.
6572:
2576:
2416:
2000:(script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and
1843:
756:
743:
504:
427:
349:
192:
11065:
10822:
10134:
9948:
8346:. It is a question of determining the correct Lagrangian density to generate the correct field equation.
423:
224:
10213:
Principles of Classical Mechanics and Field Theory / Prinzipien der Klassischen Mechanik und Feldtheorie
202:
11709:
11683:
11147:
10653:
10626:
9938:, i.e. the action (and hence the Lagrangian) does not change under a transformation parameterized by a
9935:
8335:
6507:
899:
643:
532:
458:
318:
251:
8513:
where in this context the overdot denotes a partial time derivative, not a total time derivative. The
2792:
11543:
11476:
11471:
11434:
10965:
10938:
10908:
9571:
9126:
5770:
5746:
5538:
4894:
1303:
and of parameters defining the initial positions and velocities. However, 'simple function' is not a
1285:
994:
911:
397:
8519:
6589:
5714:
5682:
2713:
is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.
11400:
11395:
11304:
11264:
10980:
10759:
10480:
10129:
8350:
7839:{\displaystyle \mathbf {a} _{k}={\ddot {\mathbf {r} }}_{k}={\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}}
7035:
5426:
1440:
Generalized coordinates incorporate constraints on the system. There is one generalized coordinate
1342:
which provide arithmetical solutions to mechanical problems to any desired degree of accuracy, the
673:
517:
2254:
1132:
11589:
10114:
9564:
7892:
7012:
2972:
1434:
1430:
1388:
1281:
1085:
864:
623:
387:
11225:
5382:
1025:
may contain an excessive amount of intricate detail that may interest only a particular audience
663:
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11584:
11558:
11380:
11375:
11370:
11365:
11323:
11286:
11232:
10958:
10948:
10926:
10843:
10705:
9119:
8887:
7501:
involve generalized accelerations, the second time derivatives of the generalized coordinates:
4513:
2572:
1362:
1308:
931:
822:
668:
485:
9235:
5526:{\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt\,.}
11511:
11491:
11429:
11358:
11137:
11115:
11090:
10990:
10848:
10817:
10680:
10601:
10094:
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9396:
3373:
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1398:
1343:
1267:
1240:
887:
678:
653:
339:
157:
10184:
10157:
9012:
first order partial differential equations, which in general will be coupled and nonlinear.
7022:
The study of the solutions of the Hamilton–Jacobi equations leads naturally to the study of
3351:
1235:
When a mechanical system contains many particles, however (such as a complex mechanism or a
11669:
11563:
11548:
11463:
11390:
11353:
11338:
11098:
11052:
10807:
10591:
10084:
9924:
8343:
8242:
7604:
as well as generalized forces mentioned above in D'Alembert's principle. The equations are
5061:
5051:
3236:
and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):
2946:
The configuration space can be defined more generally, and indeed more deeply, in terms of
2827:
2412:
1251:. In more complicated systems, the vectorial approach cannot give an adequate description.
943:
915:
879:
788:
698:
658:
566:
562:
554:
544:
334:
83:
10294:. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing.
9083:
Each transformation can be described by an operator (i.e. function acting on the position
693:
8:
11538:
11407:
11318:
11205:
11176:
10900:
10782:
10641:
10636:
10621:
10596:
10573:
10549:
10351:
10104:
10099:
9931:
9091:
variables to change them). The following are the cases when the operator does not change
7027:
7023:
5742:
5386:
3716:
1394:
1347:
1339:
1323:
1289:
1229:
1217:
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27:
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10124:
9790:
9694:
9076:
8339:
7929:
7047:
5758:
4905:
4901:
4797:
4614:
Following are overlapping properties between the Lagrangian and Hamiltonian functions.
4505:
4497:
4400:, and substituting Hamilton's equations into the result leads to the time evolution of
2843:
1997:
1358:
1190:
935:
923:
903:
868:
603:
344:
219:
187:
147:
11501:
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11215:
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11080:
10869:
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10802:
10747:
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10613:
10506:
10486:
10459:
10436:
10417:
10396:
10357:
10334:
10315:
10295:
10274:
10235:
10225:
10211:
10190:
10163:
6748:{\displaystyle \mathbf {p} ={\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }}}
5750:
4501:
3833:
2723:
2421:
The introduction of generalized coordinates and the fundamental Lagrangian function:
1406:
1402:
1277:
1255:
939:
821:
are derived from the scalar quantity by some underlying principle about the scalar's
613:
570:
527:
522:
463:
239:
229:
122:
5410:(only some are shown). The path taken by the system (red) has a stationary action (δ
4882:{\displaystyle K=H+{\frac {\partial }{\partial t}}G(\mathbf {q} ,\mathbf {p} ,t)\,,}
2846:. The general solutions form a set of possible configurations as functions of time:
2288:, and if the position vector can be written in terms of the generalized coordinates
11419:
11210:
11198:
11181:
11159:
11108:
11060:
10921:
10792:
10732:
10658:
10217:
9687:
9228:
5422:
4692:
4393:
2568:
2007:
1902:{\displaystyle \delta W={\boldsymbol {\mathcal {Q}}}\cdot \delta \mathbf {q} =0\,,}
1354:
1271:
919:
833:
814:
805:
708:
688:
633:
628:
574:
549:
404:
262:
207:
182:
9553:{\displaystyle \mathbf {p} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {p} }
9495:{\displaystyle \mathbf {r} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {r} }
11525:
11481:
11308:
11252:
11247:
11186:
11008:
10933:
10812:
10722:
10668:
10568:
10453:
9910:
8272:
4696:
3999:
2280:
1587:
703:
648:
598:
593:
512:
10505:
Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973,
10071:, which can be length, angle of rotation, or time. The corresponding momenta to
1471:
of the position space in question (usually 3 for 3d space), while the number of
11169:
10742:
10697:
10675:
10563:
10377:
7853:
particle, the second time derivative of its position vector. Each acceleration
7031:
5676:
3863:
3841:
3640:
2954:
2710:
2560:
2125:
810:
730:
638:
539:
256:
10221:
10162:(4th ed.). New York: Dover Publications Inc. Introduction, pp. xxi–xxix.
7030:. In this formulation, the solutions of the Hamilton–Jacobi equations are the
1428:
11698:
11486:
11441:
11424:
11070:
10943:
10752:
10239:
2706:
618:
445:
10517:
4482:{\displaystyle {\frac {dA}{dt}}=\{A,H\}+{\frac {\partial A}{\partial t}}\,.}
1037:
any relevant information, and removing excessive detail that may be against
11506:
11385:
11164:
11075:
10970:
10777:
8265:
8261:
8257:
7902:
7594:{\displaystyle \alpha _{r}={\ddot {q}}_{r}={\frac {d^{2}q_{r}}{dt^{2}}}\,,}
2393:
2270:
1850:
1304:
1209:
1196:
Newton's vectorial approach to mechanics describes motion with the help of
954:
882:(using generalized coordinates and corresponding generalized velocities in
837:
683:
608:
297:
177:
1326:) as in the time of Newton but most generally as a quantity determined by
1284:. This means that the analytical equations of motion do not change upon a
11617:
11533:
9914:
9015:
Again, the volume integral of the Hamiltonian density is the Hamiltonian
6691:{\displaystyle G_{2}(\mathbf {q} ,t)={\mathcal {S}}(\mathbf {q} ,t)+C\,,}
5398:
3728:
3712:
3369:
2823:
1505:
degrees of freedom, the generalized coordinates can be collected into an
891:
11553:
10797:
10787:
10435:, R.J.A. Lambourne, Open University, Cambridge University Press, 2010,
10314:
McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994,
10119:
9783:
8238:
4509:
2975:
of the Lagrangian replaces the generalized coordinates and velocities (
2830:). The particular solution to the Euler–Lagrange equations is called a
2399:
2274:
2006:
are the generalized coordinates. This leads to the generalized form of
1248:
1099: in this section. Unsourced material may be challenged and removed.
468:
142:
5735:. The path for which action is least is the path taken by the system.
3697:{\displaystyle {\mathcal {M}}=\{\mathbf {p} \in \mathbb {R} ^{N}\}\,.}
2779:{\displaystyle {\mathcal {C}}=\{\mathbf {q} \in \mathbb {R} ^{N}\}\,,}
11601:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
10916:
10648:
9939:
5536:
A general way to find the equations of motion from the action is the
3577:{\displaystyle {\frac {dH}{dt}}=-{\frac {\partial L}{\partial t}}\,,}
1468:
490:
4893:
is a frequently used letter in this case). This property is used in
4508:(indicated by hats (^)), and the Poisson bracket is replaced by the
3637:
Analogous to the configuration space, the set of all momenta is the
1074:
809:
properties of motion representing the system as a whole—usually its
5754:
5339:, equal to the total kinetic and potential energies of the system:
4609:
3627:{\displaystyle \mathbf {\dot {p}} ={\boldsymbol {\mathcal {Q}}}\,.}
2950:
2947:
2391:. For time-independent situations, the constraints are also called
1374:
on specific problems for which methods have already been designed.
1244:
1221:
1205:
872:
841:
409:
292:
267:
10395:, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,
10333:, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973,
9676:{\displaystyle \mathbf {p} \rightarrow \mathbf {p} +m\mathbf {v} }
9634:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {v} t}
6567:. If this does not hold then the transformation is not canonical.
2380:
in cases when the constraints vary with time, not just because of
1353:
Still, though lacking precise definitions, it is obvious that the
1338:. This is a fact especially at present with the modern methods of
9186:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {a} }
7001:{\displaystyle W(\mathbf {q} )={\mathcal {S}}(\mathbf {q} ,t)+Et}
949:
Analytical mechanics is used widely, from fundamental physics to
382:
235:
152:
10210:
Synge, J. L. (1960). "Classical dynamics". In Flügge, S. (ed.).
10189:(4th ed.). New York: Dover Publications Inc. pp. 3–6.
9385:{\displaystyle \mathbf {p} (t)\rightarrow \mathbf {p} (t+t_{0})}
9323:{\displaystyle \mathbf {r} (t)\rightarrow \mathbf {r} (t+t_{0})}
9070:
6808:{\displaystyle H=-{\frac {\partial {\mathcal {S}}}{\partial t}}}
4795:
Analogously, the Hamiltonian is invariant under addition of the
2279:
If the curvilinear coordinate system is defined by the standard
11656:
European Community on Computational Methods in Applied Sciences
11220:
11038:
5675:
times are fixed. The term "path" or "trajectory" refers to the
1292:
property that is lacking in the vectorial equations of motion.
1213:
441:
287:
197:
8356:
The corresponding "momentum" field densities conjugate to the
8110:
and the Euler–Lagrange equations have an analogue for fields:
5414:= 0) under small changes in the configuration of the system (δ
8249:
scalar fields, these Lagrangian field equations are a set of
1510:
1236:
1225:
1201:
845:
277:
272:
214:
11651:
International Council for Industrial and Applied Mathematics
2347:{\displaystyle \mathbf {r} =\mathbf {r} (\mathbf {q} (t),t)}
10416:
Quantum Field Theory, D. McMahon, Mc Graw Hill (US), 2008,
10273:, L.N. Hand, J.D. Finch, Cambridge University Press, 2008,
9061:{\displaystyle H=\int _{\mathcal {V}}{\mathcal {H}}\,dV\,.}
8321:{\displaystyle L=\int _{\mathcal {V}}{\mathcal {L}}\,dV\,.}
4792:. In other words, the Lagrangian of a system is not unique.
3836:
of the configuration space and generalized momentum space.
3720:
848:, of the constituents of the system; it can also be called
282:
245:
10485:. Cambridge University Press. p. §12.3, pp. 432–439.
7106:, with the rest of the coordinates non-cyclic and denoted
4599:{\displaystyle \{A,B\}\rightarrow {\frac {1}{i\hbar }}\,.}
3839:
A particular solution to Hamilton's equations is called a
3492:
first-order ordinary differential equations, one for each
1382:
9889:{\displaystyle \mathbf {p} \rightarrow -\mathbf {p} (-t)}
7898:
Generalized coordinates apply to discrete particles. For
5425:
is another quantity in analytical mechanics defined as a
1837:
75:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
16:
Overview of mechanics based on the least action principle
10067:
the Lagrangian describes the same motion independent of
9843:{\displaystyle \mathbf {r} \rightarrow \mathbf {r} (-t)}
9008:
fields, these Hamiltonian field equations are a set of 2
9004:
must be used instead of merely partial derivatives. For
4773:{\displaystyle L'=L+{\frac {d}{dt}}F(\mathbf {q} ,t)\,,}
1254:
The analytical approach simplifies problems by treating
9934:
symmetry transformation of the action corresponds to a
7862:
is expressed in terms of the generalized accelerations
4072:
is defined by the generalized coordinates and momenta:
1579:{\displaystyle \mathbf {q} =(q_{1},q_{2},\dots ,q_{N})}
875:, in which case one may revert to Newtonian mechanics.
10899:
4908:, this immediately follows from Lagrange's equations:
964:
11270:
Numerical methods for ordinary differential equations
9951:
9857:
9814:
9793:
9772:{\displaystyle \mathbf {p} \rightarrow -\mathbf {p} }
9752:
9738:{\displaystyle \mathbf {r} \rightarrow -\mathbf {r} }
9718:
9697:
9648:
9606:
9574:
9509:
9451:
9406:
9337:
9275:
9238:
9200:
9161:
9129:
9021:
8895:
8754:
8546:
8522:
8381:
8281:
8116:
7939:
7753:
7613:
7510:
7381:
7253:
7144:
6947:
6831:
6771:
6710:
6623:
6592:
6510:
6348:
5845:
5717:
5685:
5551:
5438:
5345:
5074:
5068:
the Lagrangian is explicitly time-independent, then:
4914:
4819:
4713:
4525:
4413:
4081:
4004:
All dynamical variables can be derived from position
3876:
3741:
3653:
3597:
3523:
3385:
3354:
3245:
3006:
2855:
2795:
2735:
2588:
2430:
2304:
2134:
2016:
1915:
1861:
1600:
1519:
39:
11646:
Société de Mathématiques Appliquées et Industrielles
11639:
Japan Society for Industrial and Applied Mathematics
11275:
Numerical methods for partial differential equations
10989:
10388:
10386:
9217:{\displaystyle \mathbf {p} \rightarrow \mathbf {p} }
5679:
of the system as a path through configuration space
4496:
is closely related to the equation of motion in the
1590:(here denoted by an overdot) of this tuple give the
11462:
7876:are expressed in terms the generalized coordinates
7368:Lagrangian equations in the non cyclic coordinates
6578:By setting the canonically transformed Hamiltonian
5788:) allows the Hamiltonian in one set of coordinates
1370:that include a very wide range of problems indeed.
855:Analytical mechanics takes advantage of a system's
799:is a collection of closely related formulations of
10059:
9888:
9842:
9799:
9771:
9737:
9703:
9675:
9633:
9591:
9552:
9494:
9436:
9384:
9322:
9260:
9216:
9185:
9146:
9060:
8996:
8878:
8738:
8532:
8505:
8320:
8222:
8102:
7838:
7733:
7593:
7457:
7349:
7225:
7000:
6924:
6807:
6747:
6690:
6602:
6548:
6440:
6320:
5727:
5695:
5650:
5525:
5373:
5323:
5042:
4881:
4772:
4690:The Lagrangian is invariant under addition of the
4598:
4481:
4381:
3982:
3821:
3696:
3626:
3576:
3477:
3360:
3337:
3225:
2935:
2810:
2778:
2672:
2544:
2346:
2245:
2110:
1988:
1901:
1827:
1578:
878:Two dominant branches of analytical mechanics are
74:
10412:
10410:
10408:
10383:
9530:
9472:
9419:
8256:This scalar field formulation can be extended to
7240:Hamiltonian equations for the cyclic coordinates
6582:= 0, and the type-2 generating function equal to
6501:to be canonical is the Poisson bracket be unity,
6339:such that the transformed Hamiltonian system is:
4068:) are two scalar valued dynamical variables, the
1848:D'Alembert's principle states that infinitesimal
11696:
10266:
10264:
10262:
10260:
10258:
5335:is a constant, then the Hamiltonian will be the
4610:Properties of the Lagrangian and the Hamiltonian
4504:, in which classical dynamical variables become
2571:of the entire system, then either following the
890:(using coordinates and corresponding momenta in
5764:
4687:), which would mean explicit time-independence.
11634:Society for Industrial and Applied Mathematics
10405:
9437:{\displaystyle R(\mathbf {\hat {n}} ,\theta )}
3726:The set of all positions and momenta form the
10885:
10547:
10533:
10255:
9071:Symmetry, conservation, and Noether's theorem
3719:) as used in quantum mechanics and theory of
859:to solve problems. The constraints limit the
764:
11452:Supersymmetric theory of stochastic dynamics
10478:
6537:
6511:
5393:
4538:
4526:
4449:
4437:
4117:
4104:
4098:
4086:
3963:
3877:
3812:
3772:
3687:
3664:
2916:
2856:
2769:
2746:
1318:) is not regarded as a formal expression in
10455:Mathematical methods of classical mechanics
8349:
5064:of degree 2 of the generalized velocities,
4618:All the individual generalized coordinates
2397:, for time-dependent cases they are called
1459:), i.e. each way the system can change its
1003:Learn how and when to remove these messages
10892:
10878:
10540:
10526:
7891:
2997:conjugate to the generalized coordinates:
771:
757:
10151:
10149:
9054:
9047:
8872:
8811:
8732:
8452:
8314:
8307:
8216:
8096:
7886:
7727:
7658:
7587:
7451:
7343:
7296:
7219:
7131:, they can be removed by introducing the
6684:
6434:
6389:
5644:
5519:
5367:
5317:
5283:
4952:
4948:
4875:
4766:
4592:
4475:
4371:
3976:
3959:
3939:
3935:
3922:
3815:
3799:
3690:
3677:
3620:
3570:
3471:
3426:
3219:
2929:
2912:
2892:
2888:
2878:
2798:
2772:
2759:
2726:of all possible generalized coordinates:
2666:
2575:or using the above formula – lead to the
2104:
2010:in the language of analytical mechanics:
1895:
1177:Learn how and when to remove this message
1159:Learn how and when to remove this message
1057:Learn how and when to remove this message
894:). Both formulations are equivalent by a
10376:Encyclopaedia of Physics (2nd Edition),
10283:
5397:
2960:
10451:
10349:
10290:Torby, Bruce (1984). "Energy Methods".
10186:The variational principles of mechanics
10182:
10176:
10159:The variational principles of mechanics
10155:
10140:
7444:
7414:
7207:
7197:
3615:
2991:); the generalized coordinates and the
2406:
2019:
1918:
1873:
1383:Generalized coordinates and constraints
11697:
10738:Atomic, molecular, and optical physics
10445:
10433:Relativity, Gravitation, and Cosmology
10146:
8540:is defined by analogy with mechanics:
7487:
4016:, and written as a function of these:
3993:
2354:and this relation holds for all times
1838:D'Alembert's principle of virtual work
1451:(for convenience labelled by an index
10873:
10521:
10472:
10458:(2nd ed.). Springer. Chapter 8.
10289:
10209:
7041:
6451:the above transformations are called
4671:), not simply as a parameter through
1299:are expressed as simple functions of
453:Newton's law of universal gravitation
2716:The Lagrangian formulation uses the
1212:. These quantities characterise the
1097:adding citations to reliable sources
1068:
1009:
968:
7015:for a time-independent Hamiltonian
4801:time derivative of any function of
1377:
1357:has a simple solution, whereas the
1280:, and all results are expressed in
965:Motivation for analytical mechanics
836:quantities of motion, particularly
434:Mechanics of planar particle motion
42:
13:
10901:Industrial and applied mathematics
9042:
9034:
8969:
8962:
8958:
8947:
8927:
8923:
8849:
8788:
8727:
8549:
8525:
8494:
8479:
8427:
8420:
8415:
8302:
8294:
8200:
8193:
8188:
8153:
8146:
8139:
8134:
8118:
8054:
8031:
8008:
7992:
7952:
7942:
7642:
7634:
7617:
7440:
7432:
7408:
7400:
7332:
7324:
7285:
7277:
6967:
6937:Hamilton's characteristic function
6900:
6893:
6888:
6796:
6789:
6784:
6734:
6727:
6722:
6656:
6595:
6423:
6415:
6378:
6370:
6270:
6266:
6153:
6149:
6036:
6032:
5919:
5915:
5720:
5688:
5557:
5441:
5009:
5001:
4926:
4918:
4838:
4834:
4466:
4458:
4355:
4347:
4328:
4320:
4298:
4290:
4271:
4263:
4229:
4221:
4204:
4196:
4179:
4171:
4154:
4146:
3971:
3764:
3754:
3744:
3707:("Momentum space" also refers to "
3656:
3561:
3553:
3460:
3452:
3415:
3407:
3144:
3136:
3102:
3094:
3063:
3055:
3026:
3018:
2924:
2832:(configuration) path or trajectory
2738:
2655:
2647:
2619:
2611:
2222:
2218:
2191:
2187:
2166:
2162:
2141:
2137:
2093:
2085:
2057:
2049:
1972:
1949:
1932:
14:
11721:
11131:Stochastic differential equations
6549:{\displaystyle \{Q_{i},P_{i}\}=1}
5796:to be transformed into a new set
4655:as a variable in addition to the
4553:
1397:, one customarily uses all three
984:This section has multiple issues.
11447:Supersymmetric quantum mechanics
10482:Geometric algebra for physicists
9930:Noether's theorem states that a
9870:
9859:
9824:
9816:
9765:
9754:
9731:
9720:
9669:
9658:
9650:
9624:
9616:
9608:
9582:
9546:
9527:
9511:
9488:
9469:
9453:
9416:
9356:
9339:
9294:
9277:
9210:
9202:
9179:
9171:
9163:
9137:
8709:
8682:
8622:
8396:
8083:
7808:
7774:
7756:
7712:
7336:
7305:
7289:
7258:
7189:
7181:
7165:
7162:
7152:
7013:additive separation of variables
6976:
6955:
6904:
6878:
6853:
6845:
6738:
6712:
6701:the generalized momenta become:
6665:
6638:
6427:
6400:
6397:
6382:
6355:
6352:
6301:
6293:
6248:
6240:
6217:
6209:
6184:
6176:
6131:
6123:
6100:
6092:
6067:
6059:
6014:
6006:
5983:
5975:
5950:
5942:
5897:
5889:
5866:
5858:
5617:
5614:
5604:
5498:
5495:
5485:
5308:
5305:
5295:
5276:
5173:
4862:
4854:
4753:
4233:
4208:
4183:
4158:
4130:
4122:
3901:
3884:
3787:
3779:
3668:
3604:
3601:
3464:
3437:
3434:
3419:
3392:
3389:
3320:
3317:
3307:
3291:
3288:
3278:
3261:
3253:
3035:
3032:
3008:
2860:
2811:{\displaystyle \mathbb {R} ^{N}}
2750:
2659:
2628:
2625:
2527:
2524:
2514:
2489:
2486:
2476:
2451:
2448:
2438:
2322:
2314:
2306:
2145:
2128:of the system, and the notation
2097:
2066:
2063:
1885:
1738:
1735:
1608:
1521:
1073:
1014:
973:
832:. Newtonian mechanics considers
738:
737:
724:
57:
11329:Stochastic variational calculus
11121:Ordinary differential equations
10859:Timeline of physics discoveries
10499:
10426:
10370:
10292:Advanced Dynamics for Engineers
9592:{\displaystyle G(\mathbf {v} )}
9147:{\displaystyle X(\mathbf {a} )}
8815:
8453:
7662:
7300:
6393:
5287:
3430:
2689:ordinary differential equations
1084:needs additional citations for
992:or discuss these issues on the
938:, and with some modifications,
11126:Partial differential equations
10999:Arbitrary-precision arithmetic
10356:. Vintage books. p. 474.
10343:
10324:
10308:
10246:
10203:
10054:
10051:
10045:
10027:
10021:
10015:
10006:
10003:
9991:
9973:
9961:
9955:
9883:
9874:
9863:
9837:
9828:
9820:
9758:
9724:
9654:
9612:
9586:
9578:
9542:
9521:
9515:
9484:
9463:
9457:
9431:
9410:
9379:
9360:
9352:
9349:
9343:
9317:
9298:
9290:
9287:
9281:
9255:
9242:
9206:
9167:
9141:
9133:
8988:
8965:
8719:
8705:
8692:
8678:
8632:
8554:
8533:{\displaystyle {\mathcal {H}}}
8406:
8392:
8172:
8149:
8093:
7957:
7216:
7177:
6986:
6972:
6959:
6951:
6863:
6841:
6822:is the Hamiltonian as before:
6675:
6661:
6648:
6634:
6603:{\displaystyle {\mathcal {S}}}
6311:
6289:
6258:
6236:
6227:
6205:
6194:
6172:
6141:
6119:
6110:
6088:
6077:
6055:
6024:
6002:
5993:
5971:
5960:
5938:
5907:
5885:
5876:
5854:
5753:, and is used for calculating
5728:{\displaystyle {\mathcal {C}}}
5696:{\displaystyle {\mathcal {C}}}
5629:
5600:
5510:
5481:
5314:
5291:
5280:
5222:
5199:
5196:
5177:
5166:
5119:
5107:
5081:
5078:
4949:
4872:
4850:
4763:
4749:
4589:
4583:
4568:
4559:
4541:
3914:
3911:
3905:
3894:
3888:
3880:
3791:
3775:
3332:
3303:
3271:
3249:
3216:
3177:
2870:
2864:
2539:
2510:
2501:
2472:
2463:
2434:
2341:
2332:
2326:
2318:
2260:
1983:
1926:
1819:
1747:
1573:
1528:
1:
11014:Interactive geometry software
10479:Doran, C; Lasenby, A (2003).
9913:about an axis defined by the
8748:The equations of motion are:
6610:) plus an arbitrary constant
6584:Hamilton's principal function
1334:determine the coordinates at
918:, a statement which connects
360:Koopman–von Neumann mechanics
6935:Another related function is
5765:Hamiltonian-Jacobi mechanics
1039:Knowledge's inclusion policy
803:. Analytical mechanics uses
428:Non-inertial reference frame
7:
11066:Computational number theory
11029:Numerical-analysis software
10823:Quantum information science
10183:Lanczos, Cornelius (1970).
10156:Lanczos, Cornelius (1970).
10078:
9079:in classical space and time
8275:of the Lagrangian density:
7849:is the acceleration of the
7499:Appell's equation of motion
7494:Appell's equation of motion
5060:If the kinetic energy is a
2374:is explicitly dependent on
2253:is a useful shorthand (see
1855:D'Alembert's principle is:
908:Appell's equation of motion
871:or dissipative forces like
355:Appell's equation of motion
225:Inertial frame of reference
10:
11726:
10654:Classical electromagnetism
8336:classical electromagnetism
7491:
7045:
5836:), in four possible ways:
5381:This is the basis for the
5374:{\displaystyle H=T+V=E\,.}
3997:
2964:
2842:) subject to the required
2410:
2264:
1841:
1386:
11664:
11626:
11610:
11572:
11524:
11472:Algebra of physical space
11337:
11295:
11089:
11051:
10939:Automated theorem proving
10907:
10831:
10768:
10696:
10612:
10584:
10556:
10222:10.1007/978-3-642-45943-6
7236:which leads to a set of 2
7036:Hamiltonian vector fields
7011:used to solve the HJE by
6453:canonical transformations
5771:Canonical transformations
5747:path integral formulation
5539:principle of least action
5394:Principle of least action
4895:canonical transformations
4512:of operators via Dirac's
1286:coordinate transformation
1216:of a body idealised as a
912:principle of least action
11265:Numerical linear algebra
10760:Condensed matter physics
10130:Non-autonomous mechanics
10110:Hamilton–Jacobi equation
9565:Galilean transformations
9261:{\displaystyle U(t_{0})}
9077:Symmetry transformations
8351:Hamiltonian field theory
6573:Hamilton–Jacobi equation
6331:With the restriction on
5711:) tracing out a path in
3711:-space"; the set of all
3488:which are now a set of 2
2577:Euler–Lagrange equations
2417:Euler–Lagrange equations
518:Rotating reference frame
350:Hamilton–Jacobi equation
11004:Finite element analysis
10954:Constraint satisfaction
10135:Udwadia–Kalaba equation
7893:Lagrangian field theory
7081:with conjugate momenta
5402:As the system evolves,
4902:constants of the motion
4790:exactly the same motion
2973:Legendre transformation
1495:generalized coordinates
1467:coordinates equals the
1435:generalized coordinates
1412:generalized coordinates
1405:, to refer to a body's
1389:Generalized coordinates
1282:generalized coordinates
896:Legendre transformation
865:generalized coordinates
459:Newton's laws of motion
319:Newton's laws of motion
11559:Mathematical economics
11233:Multivariable calculus
11116:Differential equations
10959:Constraint programming
10949:Computational geometry
10844:Nobel Prize in Physics
10706:Relativistic mechanics
10061:
9890:
9844:
9801:
9773:
9739:
9705:
9677:
9635:
9593:
9554:
9496:
9438:
9386:
9324:
9262:
9218:
9187:
9148:
9120:Translational symmetry
9062:
8998:
8888:variational derivative
8880:
8740:
8658:
8534:
8507:
8322:
8271:The Lagrangian is the
8224:
8104:
7887:Classical field theory
7840:
7735:
7699:
7595:
7459:
7351:
7227:
7002:
6926:
6809:
6749:
6692:
6604:
6550:
6442:
6322:
5729:
5697:
5652:
5527:
5419:
5406:traces a path through
5375:
5337:total conserved energy
5325:
5050:Such coordinates are "
5044:
4883:
4774:
4600:
4514:canonical quantization
4483:
4383:
3984:
3845:, a particular curve (
3823:
3698:
3628:
3578:
3479:
3362:
3361:{\displaystyle \cdot }
3339:
3227:
2937:
2834:, i.e. one particular
2812:
2780:
2674:
2573:calculus of variations
2546:
2348:
2247:
2112:
1990:
1903:
1844:D'Alembert's principle
1829:
1592:generalized velocities
1580:
1363:mathematical structure
1344:differential equations
1268:variational principles
1108:"Analytical mechanics"
932:relativistic mechanics
900:Hamilton–Jacobi theory
486:Simple harmonic motion
399:Euler's laws of motion
193:D'Alembert's principle
76:
11512:Supersymmetry algebra
11497:Representation theory
11492:Renormalization group
11138:Differential geometry
11019:Optimization software
10991:Mathematical software
10849:Philosophy of physics
10095:Theoretical mechanics
10090:Hamiltonian mechanics
10062:
9891:
9845:
9802:
9774:
9740:
9706:
9678:
9636:
9594:
9555:
9497:
9439:
9397:Rotational invariance
9387:
9325:
9263:
9219:
9188:
9149:
9063:
8999:
8881:
8741:
8638:
8535:
8508:
8323:
8225:
8105:
7841:
7736:
7679:
7596:
7484:equations of motion.
7460:
7352:
7228:
7003:
6927:
6810:
6750:
6693:
6605:
6551:
6443:
6323:
5738:From this principle,
5730:
5698:
5653:
5528:
5401:
5376:
5326:
5045:
4884:
4775:
4601:
4484:
4396:of one of these, say
4384:
3985:
3824:
3699:
3629:
3579:
3480:
3363:
3340:
3228:
2967:Hamiltonian mechanics
2961:Hamiltonian mechanics
2938:
2813:
2781:
2675:
2547:
2366:holonomic constraints
2349:
2267:Holonomic constraints
2248:
2113:
1991:
1904:
1830:
1581:
1399:Cartesian coordinates
1307:concept: nowadays, a
888:Hamiltonian mechanics
797:theoretical mechanics
340:Hamiltonian mechanics
158:Statistical mechanics
77:
11705:Mathematical physics
11564:Mathematical finance
11549:Social choice theory
11464:Algebraic structures
11413:in quantum mechanics
11349:Analytical mechanics
11315:Stochastic processes
11287:Variational calculus
11099:Approximation theory
11024:Statistical software
10808:Mathematical physics
10452:Arnolʹd, VI (1989).
10350:Penrose, R. (2007).
10271:Analytical Mechanics
10141:References and notes
10115:Hamilton's principle
10085:Lagrangian mechanics
9949:
9855:
9812:
9791:
9750:
9716:
9695:
9646:
9604:
9572:
9507:
9449:
9404:
9335:
9273:
9236:
9198:
9159:
9127:
9019:
8893:
8752:
8544:
8520:
8379:
8344:quantum field theory
8279:
8243:summation convention
8114:
7937:
7751:
7611:
7508:
7476:degrees of freedom.
7379:
7251:
7142:
7024:symplectic manifolds
6945:
6829:
6769:
6708:
6621:
6590:
6508:
6346:
5843:
5715:
5683:
5661:where the departure
5549:
5436:
5389:directly obtains it.
5383:Schrödinger equation
5343:
5072:
5062:homogeneous function
4912:
4817:
4711:
4523:
4411:
4079:
3874:
3739:
3717:De Broglie relations
3651:
3595:
3521:
3383:
3374:Hamilton's equations
3352:
3243:
3004:
2853:
2828:set-builder notation
2793:
2733:
2586:
2428:
2413:Lagrangian mechanics
2407:Lagrangian mechanics
2302:
2257:for this notation).
2132:
2014:
1913:
1859:
1598:
1517:
1348:difference equations
1093:improve this article
944:quantum field theory
922:to their associated
880:Lagrangian mechanics
793:analytical mechanics
789:mathematical physics
563:Angular acceleration
555:Rotational frequency
335:Lagrangian mechanics
328:Analytical mechanics
84:Second law of motion
37:
11539:Operations research
11408:Perturbation theory
11206:Multilinear algebra
11177:Functional analysis
11034:Numerical libraries
10966:Computational logic
10783:Atmospheric physics
10622:Classical mechanics
10550:branches of physics
10353:The Road to Reality
10331:Classical Mechanics
10100:Classical mechanics
10075:will be conserved.
10060:{\displaystyle L=L}
9099:, i.e. symmetries.
8515:Hamiltonian density
8245:has been used. For
7726:
7488:Appellian mechanics
7056:cyclic coordinates
7028:symplectic topology
6464:generating function
5743:equations of motion
5596:
5477:
5429:of the Lagrangian:
5408:configuration space
4780:so each Lagrangian
4699:of any function of
3994:The Poisson bracket
2994:generalized momenta
2722:of the system, the
2719:configuration space
2683:which are a set of
1429:Difference between
1395:Newtonian mechanics
1324:elementary function
1230:gravitational field
1200:quantities such as
951:applied mathematics
884:configuration space
869:conservative forces
850:vectorial mechanics
830:Newtonian mechanics
819:equations of motion
801:classical mechanics
785:theoretical physics
415:Harmonic oscillator
393:Equations of motion
28:Classical mechanics
22:Part of a series on
11676:Mathematics portal
11573:Other applications
11297:Probability theory
11280:Validated numerics
11260:Numerical analysis
11153:Geometric analysis
11143:Differential forms
10976:Information theory
10839:History of physics
10125:Kinetics (physics)
10057:
9886:
9840:
9797:
9769:
9735:
9701:
9673:
9631:
9589:
9550:
9492:
9434:
9382:
9320:
9258:
9214:
9183:
9144:
9058:
8994:
8876:
8736:
8530:
8503:
8340:general relativity
8318:
8220:
8100:
7930:Lagrangian density
7836:
7731:
7710:
7591:
7455:
7347:
7223:
7048:Routhian mechanics
7042:Routhian mechanics
6998:
6922:
6805:
6745:
6688:
6600:
6546:
6470:th kind" or "type-
6438:
6318:
6316:
5759:general relativity
5725:
5693:
5648:
5568:
5523:
5449:
5420:
5371:
5321:
5040:
4879:
4770:
4596:
4498:Heisenberg picture
4479:
4379:
4377:
4259:
3980:
3819:
3694:
3624:
3574:
3475:
3372:, also leading to
3358:
3335:
3223:
2933:
2844:initial conditions
2808:
2776:
2670:
2542:
2344:
2243:
2108:
1998:generalized forces
1986:
1899:
1825:
1576:
1501:For a system with
1491:degrees of freedom
1359:three-body problem
1346:being replaced by
1340:computer modelling
1256:mechanical systems
1191:mathematical model
936:general relativity
904:Routhian mechanics
861:degrees of freedom
731:Physics portal
345:Routhian mechanics
220:Frame of reference
72:
11710:Dynamical systems
11692:
11691:
11526:Decision sciences
11520:
11519:
11502:Spacetime algebra
11194:Harmonic analysis
11160:Dynamical systems
11104:Clifford analysis
11081:Discrete geometry
11047:
11046:
10867:
10866:
10854:Physics education
10803:Materials science
10770:Interdisciplinary
10728:Quantum mechanics
10492:978-0-521-71595-9
10465:978-0-387-96890-2
10441:978-0-521-13138-4
10422:978-0-07-154382-8
10401:978-0-13-146100-0
10393:Quantum Mechanics
10363:978-0-679-77631-4
10279:978-0-521-57572-0
10231:978-3-540-02547-4
10196:978-0-486-65067-8
10042:
9988:
9925:Noether's theorem
9899:
9898:
9800:{\displaystyle T}
9704:{\displaystyle P}
9533:
9475:
9422:
8992:
8941:
8916:
8870:
8826:
8809:
8765:
8669:
8501:
8464:
8450:
8440:
8332:Newtonian gravity
8214:
8176:
7834:
7781:
7677:
7656:
7585:
7534:
7449:
7424:
7420:
7395:
7341:
7312:
7294:
7265:
7213:
7168:
6909:
6803:
6743:
6586:(also the action
6432:
6403:
6387:
6358:
6277:
6160:
6043:
5926:
5751:quantum mechanics
5703:, in other words
5620:
5501:
5387:quantum operators
5311:
5263:
5244:
5212:
5157:
5135:
5097:
5032:
5022:
4996:
4978:
4940:
4845:
4744:
4586:
4571:
4557:
4506:quantum operators
4502:quantum mechanics
4492:This equation in
4473:
4432:
4369:
4342:
4312:
4285:
4250:
4238:
4213:
4188:
4163:
3834:Cartesian product
3607:
3568:
3542:
3469:
3440:
3424:
3395:
3323:
3294:
3167:
3157:
3125:
3115:
3086:
3076:
3042:
3038:
2664:
2635:
2631:
2602:
2530:
2492:
2454:
2236:
2205:
2180:
2150:
2102:
2073:
2069:
2040:
1810:
1782:
1760:
1741:
1722:
1686:
1656:
1621:
1449:degree of freedom
1403:coordinate system
1278:coordinate system
1187:
1186:
1179:
1169:
1168:
1161:
1143:
1067:
1066:
1059:
1007:
940:quantum mechanics
920:conservation laws
916:Noether's theorem
781:
780:
528:Centrifugal force
523:Centripetal force
479:Euler's equations
464:Relative velocity
240:Moment of inertia
70:
44:
11717:
11477:Feynman integral
11460:
11459:
11420:Potential theory
11309:random variables
11199:Fourier analysis
11182:Operator algebra
11109:Clifford algebra
11061:Computer algebra
10987:
10986:
10894:
10887:
10880:
10871:
10870:
10793:Chemical physics
10733:Particle physics
10659:Classical optics
10542:
10535:
10528:
10519:
10518:
10513:
10503:
10497:
10496:
10476:
10470:
10469:
10449:
10443:
10430:
10424:
10414:
10403:
10390:
10381:
10374:
10368:
10367:
10347:
10341:
10328:
10322:
10312:
10306:
10305:
10287:
10281:
10268:
10253:
10250:
10244:
10243:
10207:
10201:
10200:
10180:
10174:
10173:
10153:
10066:
10064:
10063:
10058:
10044:
10043:
10035:
9990:
9989:
9981:
9936:conservation law
9895:
9893:
9892:
9887:
9873:
9862:
9849:
9847:
9846:
9841:
9827:
9819:
9806:
9804:
9803:
9798:
9778:
9776:
9775:
9770:
9768:
9757:
9744:
9742:
9741:
9736:
9734:
9723:
9710:
9708:
9707:
9702:
9682:
9680:
9679:
9674:
9672:
9661:
9653:
9640:
9638:
9637:
9632:
9627:
9619:
9611:
9598:
9596:
9595:
9590:
9585:
9559:
9557:
9556:
9551:
9549:
9535:
9534:
9526:
9514:
9501:
9499:
9498:
9493:
9491:
9477:
9476:
9468:
9456:
9443:
9441:
9440:
9435:
9424:
9423:
9415:
9391:
9389:
9388:
9383:
9378:
9377:
9359:
9342:
9329:
9327:
9326:
9321:
9316:
9315:
9297:
9280:
9267:
9265:
9264:
9259:
9254:
9253:
9229:Time translation
9223:
9221:
9220:
9215:
9213:
9205:
9192:
9190:
9189:
9184:
9182:
9174:
9166:
9153:
9151:
9150:
9145:
9140:
9102:
9101:
9067:
9065:
9064:
9059:
9046:
9045:
9039:
9038:
9037:
9003:
9001:
9000:
8995:
8993:
8991:
8987:
8986:
8977:
8976:
8957:
8955:
8954:
8942:
8940:
8939:
8938:
8922:
8917:
8915:
8914:
8913:
8897:
8885:
8883:
8882:
8877:
8871:
8869:
8868:
8867:
8854:
8853:
8852:
8842:
8834:
8833:
8828:
8827:
8819:
8810:
8808:
8807:
8806:
8793:
8792:
8791:
8781:
8773:
8772:
8767:
8766:
8758:
8745:
8743:
8742:
8737:
8731:
8730:
8712:
8704:
8703:
8685:
8677:
8676:
8671:
8670:
8662:
8657:
8652:
8625:
8611:
8610:
8598:
8597:
8579:
8578:
8566:
8565:
8553:
8552:
8539:
8537:
8536:
8531:
8529:
8528:
8512:
8510:
8509:
8504:
8502:
8500:
8492:
8491:
8490:
8477:
8472:
8471:
8466:
8465:
8457:
8451:
8449:
8448:
8447:
8442:
8441:
8433:
8425:
8424:
8423:
8413:
8399:
8391:
8390:
8327:
8325:
8324:
8319:
8306:
8305:
8299:
8298:
8297:
8229:
8227:
8226:
8221:
8215:
8213:
8212:
8211:
8198:
8197:
8196:
8186:
8181:
8177:
8175:
8171:
8170:
8161:
8160:
8144:
8143:
8142:
8132:
8126:
8125:
8109:
8107:
8106:
8101:
8086:
8072:
8071:
8062:
8061:
8049:
8048:
8039:
8038:
8020:
8019:
8004:
8003:
7982:
7981:
7969:
7968:
7956:
7955:
7946:
7945:
7869:, likewise each
7845:
7843:
7842:
7837:
7835:
7833:
7832:
7831:
7818:
7817:
7816:
7811:
7805:
7804:
7794:
7789:
7788:
7783:
7782:
7777:
7772:
7765:
7764:
7759:
7740:
7738:
7737:
7732:
7725:
7720:
7715:
7709:
7708:
7698:
7693:
7678:
7670:
7657:
7655:
7654:
7653:
7640:
7632:
7627:
7626:
7621:
7620:
7600:
7598:
7597:
7592:
7586:
7584:
7583:
7582:
7569:
7568:
7567:
7558:
7557:
7547:
7542:
7541:
7536:
7535:
7527:
7520:
7519:
7479:The coordinates
7464:
7462:
7461:
7456:
7450:
7448:
7447:
7438:
7430:
7425:
7423:
7422:
7421:
7413:
7406:
7398:
7396:
7394:
7383:
7356:
7354:
7353:
7348:
7342:
7340:
7339:
7330:
7322:
7314:
7313:
7308:
7303:
7295:
7293:
7292:
7283:
7275:
7267:
7266:
7261:
7256:
7232:
7230:
7229:
7224:
7215:
7214:
7206:
7200:
7192:
7184:
7170:
7169:
7161:
7155:
7007:
7005:
7004:
6999:
6979:
6971:
6970:
6958:
6931:
6929:
6928:
6923:
6921:
6917:
6910:
6908:
6907:
6898:
6897:
6896:
6886:
6881:
6856:
6848:
6814:
6812:
6811:
6806:
6804:
6802:
6794:
6793:
6792:
6782:
6754:
6752:
6751:
6746:
6744:
6742:
6741:
6732:
6731:
6730:
6720:
6715:
6697:
6695:
6694:
6689:
6668:
6660:
6659:
6641:
6633:
6632:
6609:
6607:
6606:
6601:
6599:
6598:
6555:
6553:
6552:
6547:
6536:
6535:
6523:
6522:
6455:, each function
6447:
6445:
6444:
6439:
6433:
6431:
6430:
6421:
6413:
6405:
6404:
6396:
6388:
6386:
6385:
6376:
6368:
6360:
6359:
6351:
6327:
6325:
6324:
6319:
6317:
6304:
6296:
6288:
6287:
6278:
6276:
6265:
6251:
6243:
6220:
6212:
6200:
6187:
6179:
6171:
6170:
6161:
6159:
6148:
6134:
6126:
6103:
6095:
6083:
6070:
6062:
6054:
6053:
6044:
6042:
6031:
6017:
6009:
5986:
5978:
5966:
5953:
5945:
5937:
5936:
5927:
5925:
5914:
5900:
5892:
5869:
5861:
5849:
5734:
5732:
5731:
5726:
5724:
5723:
5702:
5700:
5699:
5694:
5692:
5691:
5657:
5655:
5654:
5649:
5622:
5621:
5613:
5607:
5595:
5594:
5593:
5583:
5582:
5581:
5561:
5560:
5532:
5530:
5529:
5524:
5503:
5502:
5494:
5488:
5476:
5475:
5474:
5464:
5463:
5462:
5445:
5444:
5380:
5378:
5377:
5372:
5330:
5328:
5327:
5322:
5313:
5312:
5304:
5298:
5279:
5271:
5270:
5265:
5264:
5256:
5252:
5251:
5246:
5245:
5237:
5230:
5229:
5220:
5219:
5214:
5213:
5205:
5192:
5191:
5176:
5165:
5164:
5159:
5158:
5150:
5143:
5142:
5137:
5136:
5128:
5115:
5114:
5105:
5104:
5099:
5098:
5090:
5049:
5047:
5046:
5041:
5033:
5031:
5030:
5029:
5024:
5023:
5015:
5007:
4999:
4997:
4995:
4984:
4979:
4977:
4969:
4968:
4967:
4954:
4941:
4939:
4938:
4937:
4924:
4916:
4888:
4886:
4885:
4880:
4865:
4857:
4846:
4844:
4833:
4779:
4777:
4776:
4771:
4756:
4745:
4743:
4732:
4721:
4605:
4603:
4602:
4597:
4588:
4587:
4579:
4573:
4572:
4564:
4558:
4556:
4545:
4488:
4486:
4485:
4480:
4474:
4472:
4464:
4456:
4433:
4431:
4423:
4415:
4394:total derivative
4392:Calculating the
4388:
4386:
4385:
4380:
4378:
4370:
4368:
4367:
4366:
4353:
4345:
4343:
4341:
4340:
4339:
4326:
4318:
4313:
4311:
4310:
4309:
4296:
4288:
4286:
4284:
4283:
4282:
4269:
4261:
4258:
4243:
4239:
4237:
4236:
4227:
4219:
4214:
4212:
4211:
4202:
4194:
4189:
4187:
4186:
4177:
4169:
4164:
4162:
4161:
4152:
4144:
4135:
4134:
4133:
4125:
3989:
3987:
3986:
3981:
3975:
3974:
3962:
3934:
3933:
3925:
3904:
3887:
3828:
3826:
3825:
3820:
3811:
3810:
3802:
3790:
3782:
3768:
3767:
3758:
3757:
3748:
3747:
3703:
3701:
3700:
3695:
3686:
3685:
3680:
3671:
3660:
3659:
3633:
3631:
3630:
3625:
3619:
3618:
3609:
3608:
3600:
3583:
3581:
3580:
3575:
3569:
3567:
3559:
3551:
3543:
3541:
3533:
3525:
3484:
3482:
3481:
3476:
3470:
3468:
3467:
3458:
3450:
3442:
3441:
3433:
3425:
3423:
3422:
3413:
3405:
3397:
3396:
3388:
3367:
3365:
3364:
3359:
3344:
3342:
3341:
3336:
3325:
3324:
3316:
3310:
3296:
3295:
3287:
3281:
3264:
3256:
3232:
3230:
3229:
3224:
3215:
3214:
3202:
3201:
3189:
3188:
3173:
3169:
3168:
3166:
3165:
3164:
3159:
3158:
3150:
3142:
3134:
3126:
3124:
3123:
3122:
3117:
3116:
3108:
3100:
3092:
3087:
3085:
3084:
3083:
3078:
3077:
3069:
3061:
3053:
3043:
3041:
3040:
3039:
3031:
3024:
3016:
3011:
2942:
2940:
2939:
2934:
2928:
2927:
2915:
2887:
2886:
2881:
2863:
2826:space (see also
2817:
2815:
2814:
2809:
2807:
2806:
2801:
2785:
2783:
2782:
2777:
2768:
2767:
2762:
2753:
2742:
2741:
2679:
2677:
2676:
2671:
2665:
2663:
2662:
2653:
2645:
2640:
2636:
2634:
2633:
2632:
2624:
2617:
2609:
2603:
2601:
2590:
2569:potential energy
2551:
2549:
2548:
2543:
2532:
2531:
2523:
2517:
2494:
2493:
2485:
2479:
2456:
2455:
2447:
2441:
2390:
2379:
2373:
2363:
2357:
2353:
2351:
2350:
2345:
2325:
2317:
2309:
2297:
2293:
2287:
2252:
2250:
2249:
2244:
2242:
2238:
2237:
2235:
2234:
2233:
2217:
2206:
2204:
2203:
2202:
2186:
2181:
2179:
2178:
2177:
2161:
2151:
2149:
2148:
2136:
2117:
2115:
2114:
2109:
2103:
2101:
2100:
2091:
2083:
2078:
2074:
2072:
2071:
2070:
2062:
2055:
2047:
2041:
2039:
2028:
2023:
2022:
2005:
1995:
1993:
1992:
1987:
1982:
1981:
1976:
1975:
1959:
1958:
1953:
1952:
1942:
1941:
1936:
1935:
1922:
1921:
1908:
1906:
1905:
1900:
1888:
1877:
1876:
1834:
1832:
1831:
1826:
1818:
1817:
1812:
1811:
1803:
1790:
1789:
1784:
1783:
1775:
1768:
1767:
1762:
1761:
1753:
1743:
1742:
1734:
1728:
1724:
1723:
1721:
1713:
1712:
1711:
1698:
1687:
1685:
1677:
1676:
1675:
1662:
1657:
1655:
1647:
1646:
1645:
1632:
1622:
1620:
1612:
1611:
1602:
1585:
1583:
1582:
1577:
1572:
1571:
1553:
1552:
1540:
1539:
1524:
1480:× − (number of
1378:Intrinsic motion
1355:two-body problem
1182:
1175:
1164:
1157:
1153:
1150:
1144:
1142:
1101:
1077:
1069:
1062:
1055:
1051:
1048:
1042:
1018:
1017:
1010:
999:
977:
976:
969:
914:. One result is
815:potential energy
773:
766:
759:
746:
741:
740:
733:
729:
728:
634:Johann Bernoulli
629:Daniel Bernoulli
550:Tangential speed
454:
430:
405:Fictitious force
400:
252:Mechanical power
242:
183:Angular momentum
81:
79:
78:
73:
71:
69:
61:
60:
51:
46:
45:
19:
18:
11725:
11724:
11720:
11719:
11718:
11716:
11715:
11714:
11695:
11694:
11693:
11688:
11660:
11622:
11606:
11568:
11516:
11482:Poisson algebra
11458:
11340:
11333:
11291:
11187:Operator theory
11085:
11043:
11009:Tensor software
10985:
10934:Automata theory
10903:
10898:
10868:
10863:
10827:
10813:Medical physics
10764:
10723:Nuclear physics
10692:
10686:Non-equilibrium
10608:
10580:
10552:
10546:
10516:
10504:
10500:
10493:
10477:
10473:
10466:
10450:
10446:
10431:
10427:
10415:
10406:
10391:
10384:
10375:
10371:
10364:
10348:
10344:
10329:
10325:
10313:
10309:
10302:
10288:
10284:
10269:
10256:
10251:
10247:
10232:
10208:
10204:
10197:
10181:
10177:
10170:
10154:
10147:
10143:
10081:
10034:
10033:
9980:
9979:
9950:
9947:
9946:
9911:rotation matrix
9869:
9858:
9856:
9853:
9852:
9823:
9815:
9813:
9810:
9809:
9792:
9789:
9788:
9764:
9753:
9751:
9748:
9747:
9730:
9719:
9717:
9714:
9713:
9696:
9693:
9692:
9668:
9657:
9649:
9647:
9644:
9643:
9623:
9615:
9607:
9605:
9602:
9601:
9581:
9573:
9570:
9569:
9545:
9525:
9524:
9510:
9508:
9505:
9504:
9487:
9467:
9466:
9452:
9450:
9447:
9446:
9414:
9413:
9405:
9402:
9401:
9373:
9369:
9355:
9338:
9336:
9333:
9332:
9311:
9307:
9293:
9276:
9274:
9271:
9270:
9249:
9245:
9237:
9234:
9233:
9209:
9201:
9199:
9196:
9195:
9178:
9170:
9162:
9160:
9157:
9156:
9136:
9128:
9125:
9124:
9105:Transformation
9073:
9041:
9040:
9033:
9032:
9028:
9020:
9017:
9016:
8982:
8978:
8972:
8968:
8961:
8956:
8950:
8946:
8934:
8930:
8926:
8921:
8909:
8905:
8901:
8896:
8894:
8891:
8890:
8863:
8859:
8855:
8848:
8847:
8843:
8841:
8829:
8818:
8817:
8816:
8802:
8798:
8794:
8787:
8786:
8782:
8780:
8768:
8757:
8756:
8755:
8753:
8750:
8749:
8726:
8725:
8708:
8699:
8695:
8681:
8672:
8661:
8660:
8659:
8653:
8642:
8621:
8606:
8602:
8593:
8589:
8574:
8570:
8561:
8557:
8548:
8547:
8545:
8542:
8541:
8524:
8523:
8521:
8518:
8517:
8493:
8486:
8482:
8478:
8476:
8467:
8456:
8455:
8454:
8443:
8432:
8431:
8430:
8426:
8419:
8418:
8414:
8412:
8395:
8386:
8382:
8380:
8377:
8376:
8365:
8354:
8301:
8300:
8293:
8292:
8288:
8280:
8277:
8276:
8273:volume integral
8235:
8207:
8203:
8199:
8192:
8191:
8187:
8185:
8166:
8162:
8156:
8152:
8145:
8138:
8137:
8133:
8131:
8127:
8121:
8117:
8115:
8112:
8111:
8082:
8067:
8063:
8057:
8053:
8044:
8040:
8034:
8030:
8015:
8011:
7999:
7995:
7977:
7973:
7964:
7960:
7951:
7950:
7941:
7940:
7938:
7935:
7934:
7909:
7896:
7889:
7881:
7875:
7867:
7861:
7827:
7823:
7819:
7812:
7807:
7806:
7800:
7796:
7795:
7793:
7784:
7773:
7771:
7770:
7769:
7760:
7755:
7754:
7752:
7749:
7748:
7721:
7716:
7711:
7704:
7700:
7694:
7683:
7669:
7649:
7645:
7641:
7633:
7631:
7622:
7616:
7615:
7614:
7612:
7609:
7608:
7578:
7574:
7570:
7563:
7559:
7553:
7549:
7548:
7546:
7537:
7526:
7525:
7524:
7515:
7511:
7509:
7506:
7505:
7496:
7490:
7443:
7439:
7431:
7429:
7412:
7411:
7407:
7399:
7397:
7387:
7382:
7380:
7377:
7376:
7335:
7331:
7323:
7321:
7304:
7302:
7301:
7288:
7284:
7276:
7274:
7257:
7255:
7254:
7252:
7249:
7248:
7205:
7204:
7196:
7188:
7180:
7160:
7159:
7151:
7143:
7140:
7139:
7129:
7123:
7116:
7104:
7098:
7091:
7079:
7073:
7066:
7050:
7044:
7032:integral curves
6975:
6966:
6965:
6954:
6946:
6943:
6942:
6903:
6899:
6892:
6891:
6887:
6885:
6877:
6876:
6872:
6852:
6844:
6830:
6827:
6826:
6795:
6788:
6787:
6783:
6781:
6770:
6767:
6766:
6737:
6733:
6726:
6725:
6721:
6719:
6711:
6709:
6706:
6705:
6664:
6655:
6654:
6637:
6628:
6624:
6622:
6619:
6618:
6594:
6593:
6591:
6588:
6587:
6531:
6527:
6518:
6514:
6509:
6506:
6505:
6460:
6426:
6422:
6414:
6412:
6395:
6394:
6381:
6377:
6369:
6367:
6350:
6349:
6347:
6344:
6343:
6315:
6314:
6300:
6292:
6283:
6279:
6269:
6264:
6247:
6239:
6216:
6208:
6198:
6197:
6183:
6175:
6166:
6162:
6152:
6147:
6130:
6122:
6099:
6091:
6081:
6080:
6066:
6058:
6049:
6045:
6035:
6030:
6013:
6005:
5982:
5974:
5964:
5963:
5949:
5941:
5932:
5928:
5918:
5913:
5896:
5888:
5865:
5857:
5846:
5844:
5841:
5840:
5767:
5719:
5718:
5716:
5713:
5712:
5687:
5686:
5684:
5681:
5680:
5674:
5667:
5612:
5611:
5603:
5589:
5585:
5584:
5577:
5573:
5572:
5556:
5555:
5550:
5547:
5546:
5493:
5492:
5484:
5470:
5466:
5465:
5458:
5454:
5453:
5440:
5439:
5437:
5434:
5433:
5396:
5344:
5341:
5340:
5303:
5302:
5294:
5275:
5266:
5255:
5254:
5253:
5247:
5236:
5235:
5234:
5225:
5221:
5215:
5204:
5203:
5202:
5187:
5183:
5172:
5160:
5149:
5148:
5147:
5138:
5127:
5126:
5125:
5110:
5106:
5100:
5089:
5088:
5087:
5073:
5070:
5069:
5025:
5014:
5013:
5012:
5008:
5000:
4998:
4988:
4983:
4970:
4963:
4959:
4955:
4953:
4933:
4929:
4925:
4917:
4915:
4913:
4910:
4909:
4861:
4853:
4837:
4832:
4818:
4815:
4814:
4752:
4736:
4731:
4714:
4712:
4709:
4708:
4697:time derivative
4645:
4634:
4623:
4612:
4578:
4577:
4563:
4562:
4549:
4544:
4524:
4521:
4520:
4465:
4457:
4455:
4424:
4416:
4414:
4412:
4409:
4408:
4376:
4375:
4362:
4358:
4354:
4346:
4344:
4335:
4331:
4327:
4319:
4317:
4305:
4301:
4297:
4289:
4287:
4278:
4274:
4270:
4262:
4260:
4254:
4241:
4240:
4232:
4228:
4220:
4218:
4207:
4203:
4195:
4193:
4182:
4178:
4170:
4168:
4157:
4153:
4145:
4143:
4136:
4129:
4121:
4120:
4116:
4082:
4080:
4077:
4076:
4070:Poisson bracket
4002:
4000:Poisson bracket
3996:
3970:
3969:
3958:
3926:
3921:
3920:
3900:
3883:
3875:
3872:
3871:
3803:
3798:
3797:
3786:
3778:
3763:
3762:
3753:
3752:
3743:
3742:
3740:
3737:
3736:
3681:
3676:
3675:
3667:
3655:
3654:
3652:
3649:
3648:
3614:
3613:
3599:
3598:
3596:
3593:
3592:
3560:
3552:
3550:
3534:
3526:
3524:
3522:
3519:
3518:
3508:
3497:
3463:
3459:
3451:
3449:
3432:
3431:
3418:
3414:
3406:
3404:
3387:
3386:
3384:
3381:
3380:
3353:
3350:
3349:
3315:
3314:
3306:
3286:
3285:
3277:
3260:
3252:
3244:
3241:
3240:
3210:
3206:
3197:
3193:
3184:
3180:
3160:
3149:
3148:
3147:
3143:
3135:
3133:
3118:
3107:
3106:
3105:
3101:
3093:
3091:
3079:
3068:
3067:
3066:
3062:
3054:
3052:
3051:
3047:
3030:
3029:
3025:
3017:
3015:
3007:
3005:
3002:
3001:
2969:
2963:
2923:
2922:
2911:
2882:
2877:
2876:
2859:
2854:
2851:
2850:
2802:
2797:
2796:
2794:
2791:
2790:
2763:
2758:
2757:
2749:
2737:
2736:
2734:
2731:
2730:
2696:
2691:, one for each
2658:
2654:
2646:
2644:
2623:
2622:
2618:
2610:
2608:
2604:
2594:
2589:
2587:
2584:
2583:
2522:
2521:
2513:
2484:
2483:
2475:
2446:
2445:
2437:
2429:
2426:
2425:
2419:
2411:Main articles:
2409:
2381:
2375:
2369:
2359:
2355:
2321:
2313:
2305:
2303:
2300:
2299:
2295:
2289:
2283:
2281:position vector
2277:
2265:Main articles:
2263:
2255:matrix calculus
2229:
2225:
2221:
2216:
2198:
2194:
2190:
2185:
2173:
2169:
2165:
2160:
2159:
2155:
2144:
2140:
2135:
2133:
2130:
2129:
2096:
2092:
2084:
2082:
2061:
2060:
2056:
2048:
2046:
2042:
2032:
2027:
2018:
2017:
2015:
2012:
2011:
2001:
1977:
1971:
1970:
1969:
1954:
1948:
1947:
1946:
1937:
1931:
1930:
1929:
1917:
1916:
1914:
1911:
1910:
1884:
1872:
1871:
1860:
1857:
1856:
1846:
1840:
1813:
1802:
1801:
1800:
1785:
1774:
1773:
1772:
1763:
1752:
1751:
1750:
1733:
1732:
1714:
1707:
1703:
1699:
1697:
1678:
1671:
1667:
1663:
1661:
1648:
1641:
1637:
1633:
1631:
1630:
1626:
1613:
1607:
1603:
1601:
1599:
1596:
1595:
1588:time derivative
1567:
1563:
1548:
1544:
1535:
1531:
1520:
1518:
1515:
1514:
1499:
1493:) = (number of
1486:
1445:
1438:
1425:= 1, 2, 3...).
1419:
1391:
1385:
1380:
1183:
1172:
1171:
1170:
1165:
1154:
1148:
1145:
1102:
1100:
1090:
1078:
1063:
1052:
1046:
1043:
1029:Please help by
1028:
1019:
1015:
978:
974:
967:
953:, particularly
777:
736:
723:
722:
715:
714:
713:
588:
580:
579:
559:
513:Circular motion
507:
497:
496:
495:
452:
422:
419:
398:
377:
369:
368:
365:
364:
322:
312:
304:
303:
302:
261:
257:Mechanical work
250:
234:
172:
164:
163:
162:
117:
109:
86:
62:
56:
52:
50:
41:
40:
38:
35:
34:
17:
12:
11:
5:
11723:
11713:
11712:
11707:
11690:
11689:
11687:
11686:
11673:
11665:
11662:
11661:
11659:
11658:
11653:
11648:
11643:
11642:
11641:
11630:
11628:
11624:
11623:
11621:
11620:
11614:
11612:
11608:
11607:
11605:
11604:
11597:
11592:
11587:
11582:
11576:
11574:
11570:
11569:
11567:
11566:
11561:
11556:
11551:
11546:
11541:
11536:
11530:
11528:
11522:
11521:
11518:
11517:
11515:
11514:
11509:
11504:
11499:
11494:
11489:
11484:
11479:
11474:
11468:
11466:
11457:
11456:
11455:
11454:
11449:
11439:
11438:
11437:
11432:
11422:
11417:
11416:
11415:
11405:
11404:
11403:
11398:
11393:
11388:
11383:
11378:
11373:
11363:
11362:
11361:
11356:
11345:
11343:
11335:
11334:
11332:
11331:
11326:
11321:
11312:
11301:
11299:
11293:
11292:
11290:
11289:
11284:
11283:
11282:
11277:
11272:
11267:
11257:
11256:
11255:
11250:
11245:
11240:
11230:
11229:
11228:
11223:
11218:
11213:
11203:
11202:
11201:
11191:
11190:
11189:
11184:
11174:
11173:
11172:
11170:Control theory
11167:
11157:
11156:
11155:
11150:
11145:
11135:
11134:
11133:
11128:
11123:
11113:
11112:
11111:
11101:
11095:
11093:
11087:
11086:
11084:
11083:
11078:
11073:
11068:
11063:
11057:
11055:
11049:
11048:
11045:
11044:
11042:
11041:
11036:
11031:
11026:
11021:
11016:
11011:
11006:
11001:
10995:
10993:
10984:
10983:
10978:
10973:
10968:
10963:
10962:
10961:
10951:
10946:
10941:
10936:
10931:
10930:
10929:
10924:
10913:
10911:
10905:
10904:
10897:
10896:
10889:
10882:
10874:
10865:
10864:
10862:
10861:
10856:
10851:
10846:
10841:
10835:
10833:
10829:
10828:
10826:
10825:
10820:
10815:
10810:
10805:
10800:
10795:
10790:
10785:
10780:
10774:
10772:
10766:
10765:
10763:
10762:
10757:
10756:
10755:
10750:
10745:
10735:
10730:
10725:
10720:
10719:
10718:
10713:
10702:
10700:
10694:
10693:
10691:
10690:
10689:
10688:
10683:
10676:Thermodynamics
10673:
10672:
10671:
10666:
10656:
10651:
10646:
10645:
10644:
10639:
10634:
10629:
10618:
10616:
10610:
10609:
10607:
10606:
10605:
10604:
10594:
10588:
10586:
10582:
10581:
10579:
10578:
10577:
10576:
10566:
10560:
10558:
10554:
10553:
10545:
10544:
10537:
10530:
10522:
10515:
10514:
10498:
10491:
10471:
10464:
10444:
10425:
10404:
10382:
10369:
10362:
10342:
10323:
10307:
10300:
10282:
10254:
10245:
10230:
10202:
10195:
10175:
10168:
10144:
10142:
10139:
10138:
10137:
10132:
10127:
10122:
10117:
10112:
10107:
10102:
10097:
10092:
10087:
10080:
10077:
10056:
10053:
10050:
10047:
10041:
10038:
10032:
10029:
10026:
10023:
10020:
10017:
10014:
10011:
10008:
10005:
10002:
9999:
9996:
9993:
9987:
9984:
9978:
9975:
9972:
9969:
9966:
9963:
9960:
9957:
9954:
9928:
9927:
9897:
9896:
9885:
9882:
9879:
9876:
9872:
9868:
9865:
9861:
9850:
9839:
9836:
9833:
9830:
9826:
9822:
9818:
9807:
9796:
9786:
9780:
9779:
9767:
9763:
9760:
9756:
9745:
9733:
9729:
9726:
9722:
9711:
9700:
9690:
9684:
9683:
9671:
9667:
9664:
9660:
9656:
9652:
9641:
9630:
9626:
9622:
9618:
9614:
9610:
9599:
9588:
9584:
9580:
9577:
9567:
9561:
9560:
9548:
9544:
9541:
9538:
9532:
9529:
9523:
9520:
9517:
9513:
9502:
9490:
9486:
9483:
9480:
9474:
9471:
9465:
9462:
9459:
9455:
9444:
9433:
9430:
9427:
9421:
9418:
9412:
9409:
9399:
9393:
9392:
9381:
9376:
9372:
9368:
9365:
9362:
9358:
9354:
9351:
9348:
9345:
9341:
9330:
9319:
9314:
9310:
9306:
9303:
9300:
9296:
9292:
9289:
9286:
9283:
9279:
9268:
9257:
9252:
9248:
9244:
9241:
9231:
9225:
9224:
9212:
9208:
9204:
9193:
9181:
9177:
9173:
9169:
9165:
9154:
9143:
9139:
9135:
9132:
9122:
9116:
9115:
9112:
9109:
9106:
9081:
9080:
9072:
9069:
9057:
9053:
9050:
9044:
9036:
9031:
9027:
9024:
8990:
8985:
8981:
8975:
8971:
8967:
8964:
8960:
8953:
8949:
8945:
8937:
8933:
8929:
8925:
8920:
8912:
8908:
8904:
8900:
8875:
8866:
8862:
8858:
8851:
8846:
8840:
8837:
8832:
8825:
8822:
8814:
8805:
8801:
8797:
8790:
8785:
8779:
8776:
8771:
8764:
8761:
8735:
8729:
8724:
8721:
8718:
8715:
8711:
8707:
8702:
8698:
8694:
8691:
8688:
8684:
8680:
8675:
8668:
8665:
8656:
8651:
8648:
8645:
8641:
8637:
8634:
8631:
8628:
8624:
8620:
8617:
8614:
8609:
8605:
8601:
8596:
8592:
8588:
8585:
8582:
8577:
8573:
8569:
8564:
8560:
8556:
8551:
8527:
8499:
8496:
8489:
8485:
8481:
8475:
8470:
8463:
8460:
8446:
8439:
8436:
8429:
8422:
8417:
8411:
8408:
8405:
8402:
8398:
8394:
8389:
8385:
8363:
8360:scalar fields
8353:
8348:
8317:
8313:
8310:
8304:
8296:
8291:
8287:
8284:
8233:
8219:
8210:
8206:
8202:
8195:
8190:
8184:
8180:
8174:
8169:
8165:
8159:
8155:
8151:
8148:
8141:
8136:
8130:
8124:
8120:
8099:
8095:
8092:
8089:
8085:
8081:
8078:
8075:
8070:
8066:
8060:
8056:
8052:
8047:
8043:
8037:
8033:
8029:
8026:
8023:
8018:
8014:
8010:
8007:
8002:
7998:
7994:
7991:
7988:
7985:
7980:
7976:
7972:
7967:
7963:
7959:
7954:
7949:
7944:
7907:
7895:
7890:
7888:
7885:
7879:
7873:
7865:
7857:
7847:
7846:
7830:
7826:
7822:
7815:
7810:
7803:
7799:
7792:
7787:
7780:
7776:
7768:
7763:
7758:
7742:
7741:
7730:
7724:
7719:
7714:
7707:
7703:
7697:
7692:
7689:
7686:
7682:
7676:
7673:
7668:
7665:
7661:
7652:
7648:
7644:
7639:
7636:
7630:
7625:
7619:
7602:
7601:
7590:
7581:
7577:
7573:
7566:
7562:
7556:
7552:
7545:
7540:
7533:
7530:
7523:
7518:
7514:
7492:Main article:
7489:
7486:
7466:
7465:
7454:
7446:
7442:
7437:
7434:
7428:
7419:
7416:
7410:
7405:
7402:
7393:
7390:
7386:
7358:
7357:
7346:
7338:
7334:
7329:
7326:
7320:
7317:
7311:
7307:
7299:
7291:
7287:
7282:
7279:
7273:
7270:
7264:
7260:
7234:
7233:
7222:
7218:
7212:
7209:
7203:
7199:
7195:
7191:
7187:
7183:
7179:
7176:
7173:
7167:
7164:
7158:
7154:
7150:
7147:
7127:
7121:
7114:
7102:
7096:
7089:
7077:
7071:
7064:
7046:Main article:
7043:
7040:
7009:
7008:
6997:
6994:
6991:
6988:
6985:
6982:
6978:
6974:
6969:
6964:
6961:
6957:
6953:
6950:
6933:
6932:
6920:
6916:
6913:
6906:
6902:
6895:
6890:
6884:
6880:
6875:
6871:
6868:
6865:
6862:
6859:
6855:
6851:
6847:
6843:
6840:
6837:
6834:
6816:
6815:
6801:
6798:
6791:
6786:
6780:
6777:
6774:
6756:
6755:
6740:
6736:
6729:
6724:
6718:
6714:
6699:
6698:
6687:
6683:
6680:
6677:
6674:
6671:
6667:
6663:
6658:
6653:
6650:
6647:
6644:
6640:
6636:
6631:
6627:
6597:
6576:
6575:
6557:
6556:
6545:
6542:
6539:
6534:
6530:
6526:
6521:
6517:
6513:
6477:The choice of
6458:
6449:
6448:
6437:
6429:
6425:
6420:
6417:
6411:
6408:
6402:
6399:
6392:
6384:
6380:
6375:
6372:
6366:
6363:
6357:
6354:
6329:
6328:
6313:
6310:
6307:
6303:
6299:
6295:
6291:
6286:
6282:
6275:
6272:
6268:
6263:
6260:
6257:
6254:
6250:
6246:
6242:
6238:
6235:
6232:
6229:
6226:
6223:
6219:
6215:
6211:
6207:
6204:
6201:
6199:
6196:
6193:
6190:
6186:
6182:
6178:
6174:
6169:
6165:
6158:
6155:
6151:
6146:
6143:
6140:
6137:
6133:
6129:
6125:
6121:
6118:
6115:
6112:
6109:
6106:
6102:
6098:
6094:
6090:
6087:
6084:
6082:
6079:
6076:
6073:
6069:
6065:
6061:
6057:
6052:
6048:
6041:
6038:
6034:
6029:
6026:
6023:
6020:
6016:
6012:
6008:
6004:
6001:
5998:
5995:
5992:
5989:
5985:
5981:
5977:
5973:
5970:
5967:
5965:
5962:
5959:
5956:
5952:
5948:
5944:
5940:
5935:
5931:
5924:
5921:
5917:
5912:
5909:
5906:
5903:
5899:
5895:
5891:
5887:
5884:
5881:
5878:
5875:
5872:
5868:
5864:
5860:
5856:
5853:
5850:
5848:
5774:
5773:
5766:
5763:
5722:
5690:
5677:time evolution
5672:
5665:
5659:
5658:
5647:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5619:
5616:
5610:
5606:
5602:
5599:
5592:
5588:
5580:
5576:
5571:
5567:
5564:
5559:
5554:
5534:
5533:
5522:
5518:
5515:
5512:
5509:
5506:
5500:
5497:
5491:
5487:
5483:
5480:
5473:
5469:
5461:
5457:
5452:
5448:
5443:
5395:
5392:
5391:
5390:
5370:
5366:
5363:
5360:
5357:
5354:
5351:
5348:
5320:
5316:
5310:
5307:
5301:
5297:
5293:
5290:
5286:
5282:
5278:
5274:
5269:
5262:
5259:
5250:
5243:
5240:
5233:
5228:
5224:
5218:
5211:
5208:
5201:
5198:
5195:
5190:
5186:
5182:
5179:
5175:
5171:
5168:
5163:
5156:
5153:
5146:
5141:
5134:
5131:
5124:
5121:
5118:
5113:
5109:
5103:
5096:
5093:
5086:
5083:
5080:
5077:
5058:
5055:
5039:
5036:
5028:
5021:
5018:
5011:
5006:
5003:
4994:
4991:
4987:
4982:
4976:
4973:
4966:
4962:
4958:
4951:
4947:
4944:
4936:
4932:
4928:
4923:
4920:
4898:
4878:
4874:
4871:
4868:
4864:
4860:
4856:
4852:
4849:
4843:
4840:
4836:
4831:
4828:
4825:
4822:
4793:
4769:
4765:
4762:
4759:
4755:
4751:
4748:
4742:
4739:
4735:
4730:
4727:
4724:
4720:
4717:
4688:
4643:
4640:) and momenta
4632:
4629:), velocities
4621:
4611:
4608:
4607:
4606:
4595:
4591:
4585:
4582:
4576:
4570:
4567:
4561:
4555:
4552:
4548:
4543:
4540:
4537:
4534:
4531:
4528:
4490:
4489:
4478:
4471:
4468:
4463:
4460:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4430:
4427:
4422:
4419:
4390:
4389:
4374:
4365:
4361:
4357:
4352:
4349:
4338:
4334:
4330:
4325:
4322:
4316:
4308:
4304:
4300:
4295:
4292:
4281:
4277:
4273:
4268:
4265:
4257:
4253:
4249:
4246:
4244:
4242:
4235:
4231:
4226:
4223:
4217:
4210:
4206:
4201:
4198:
4192:
4185:
4181:
4176:
4173:
4167:
4160:
4156:
4151:
4148:
4142:
4139:
4137:
4132:
4128:
4124:
4119:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4084:
3998:Main article:
3995:
3992:
3991:
3990:
3979:
3973:
3968:
3965:
3961:
3957:
3954:
3951:
3948:
3945:
3942:
3938:
3932:
3929:
3924:
3919:
3916:
3913:
3910:
3907:
3903:
3899:
3896:
3893:
3890:
3886:
3882:
3879:
3864:phase portrait
3830:
3829:
3818:
3814:
3809:
3806:
3801:
3796:
3793:
3789:
3785:
3781:
3777:
3774:
3771:
3766:
3761:
3756:
3751:
3746:
3705:
3704:
3693:
3689:
3684:
3679:
3674:
3670:
3666:
3663:
3658:
3641:momentum space
3635:
3634:
3623:
3617:
3612:
3606:
3603:
3585:
3584:
3573:
3566:
3563:
3558:
3555:
3549:
3546:
3540:
3537:
3532:
3529:
3506:
3495:
3486:
3485:
3474:
3466:
3462:
3457:
3454:
3448:
3445:
3439:
3436:
3429:
3421:
3417:
3412:
3409:
3403:
3400:
3394:
3391:
3357:
3346:
3345:
3334:
3331:
3328:
3322:
3319:
3313:
3309:
3305:
3302:
3299:
3293:
3290:
3284:
3280:
3276:
3273:
3270:
3267:
3263:
3259:
3255:
3251:
3248:
3234:
3233:
3222:
3218:
3213:
3209:
3205:
3200:
3196:
3192:
3187:
3183:
3179:
3176:
3172:
3163:
3156:
3153:
3146:
3141:
3138:
3132:
3129:
3121:
3114:
3111:
3104:
3099:
3096:
3090:
3082:
3075:
3072:
3065:
3060:
3057:
3050:
3046:
3037:
3034:
3028:
3023:
3020:
3014:
3010:
2965:Main article:
2962:
2959:
2955:tangent bundle
2944:
2943:
2932:
2926:
2921:
2918:
2914:
2910:
2907:
2904:
2901:
2898:
2895:
2891:
2885:
2880:
2875:
2872:
2869:
2866:
2862:
2858:
2805:
2800:
2787:
2786:
2775:
2771:
2766:
2761:
2756:
2752:
2748:
2745:
2740:
2711:kinetic energy
2694:
2681:
2680:
2669:
2661:
2657:
2652:
2649:
2643:
2639:
2630:
2627:
2621:
2616:
2613:
2607:
2600:
2597:
2593:
2561:kinetic energy
2553:
2552:
2541:
2538:
2535:
2529:
2526:
2520:
2516:
2512:
2509:
2506:
2503:
2500:
2497:
2491:
2488:
2482:
2478:
2474:
2471:
2468:
2465:
2462:
2459:
2453:
2450:
2444:
2440:
2436:
2433:
2408:
2405:
2343:
2340:
2337:
2334:
2331:
2328:
2324:
2320:
2316:
2312:
2308:
2262:
2259:
2241:
2232:
2228:
2224:
2220:
2215:
2212:
2209:
2201:
2197:
2193:
2189:
2184:
2176:
2172:
2168:
2164:
2158:
2154:
2147:
2143:
2139:
2126:kinetic energy
2107:
2099:
2095:
2090:
2087:
2081:
2077:
2068:
2065:
2059:
2054:
2051:
2045:
2038:
2035:
2031:
2026:
2021:
1985:
1980:
1974:
1968:
1965:
1962:
1957:
1951:
1945:
1940:
1934:
1928:
1925:
1920:
1898:
1894:
1891:
1887:
1883:
1880:
1875:
1870:
1867:
1864:
1842:Main article:
1839:
1836:
1824:
1821:
1816:
1809:
1806:
1799:
1796:
1793:
1788:
1781:
1778:
1771:
1766:
1759:
1756:
1749:
1746:
1740:
1737:
1731:
1727:
1720:
1717:
1710:
1706:
1702:
1696:
1693:
1690:
1684:
1681:
1674:
1670:
1666:
1660:
1654:
1651:
1644:
1640:
1636:
1629:
1625:
1619:
1616:
1610:
1606:
1575:
1570:
1566:
1562:
1559:
1556:
1551:
1547:
1543:
1538:
1534:
1530:
1527:
1523:
1487:
1478:
1443:
1437:
1427:
1417:
1401:, or other 3D
1387:Main article:
1384:
1381:
1379:
1376:
1185:
1184:
1167:
1166:
1081:
1079:
1072:
1065:
1064:
1022:
1020:
1013:
1008:
982:
981:
979:
972:
966:
963:
811:kinetic energy
779:
778:
776:
775:
768:
761:
753:
750:
749:
748:
747:
734:
717:
716:
712:
711:
706:
701:
696:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
621:
616:
611:
606:
601:
596:
590:
589:
586:
585:
582:
581:
578:
577:
558:
557:
552:
547:
542:
540:Coriolis force
537:
536:
535:
525:
520:
515:
509:
508:
503:
502:
499:
498:
494:
493:
488:
483:
482:
481:
476:
466:
461:
456:
449:
438:
437:
436:
431:
418:
417:
412:
407:
402:
395:
390:
385:
379:
378:
375:
374:
371:
370:
367:
366:
363:
362:
357:
352:
347:
342:
337:
331:
325:
323:
316:
313:
310:
309:
306:
305:
301:
300:
295:
290:
285:
280:
275:
270:
265:
259:
254:
248:
243:
232:
227:
222:
217:
212:
211:
210:
205:
195:
190:
185:
180:
174:
173:
170:
169:
166:
165:
161:
160:
155:
150:
145:
140:
135:
130:
125:
119:
118:
115:
114:
111:
110:
108:
107:
102:
97:
91:
88:
87:
82:
68:
65:
59:
55:
49:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
11722:
11711:
11708:
11706:
11703:
11702:
11700:
11685:
11681:
11677:
11674:
11672:
11671:
11667:
11666:
11663:
11657:
11654:
11652:
11649:
11647:
11644:
11640:
11637:
11636:
11635:
11632:
11631:
11629:
11627:Organizations
11625:
11619:
11616:
11615:
11613:
11609:
11602:
11598:
11596:
11593:
11591:
11588:
11586:
11583:
11581:
11578:
11577:
11575:
11571:
11565:
11562:
11560:
11557:
11555:
11552:
11550:
11547:
11545:
11542:
11540:
11537:
11535:
11532:
11531:
11529:
11527:
11523:
11513:
11510:
11508:
11505:
11503:
11500:
11498:
11495:
11493:
11490:
11488:
11487:Quantum group
11485:
11483:
11480:
11478:
11475:
11473:
11470:
11469:
11467:
11465:
11461:
11453:
11450:
11448:
11445:
11444:
11443:
11442:Supersymmetry
11440:
11436:
11433:
11431:
11428:
11427:
11426:
11425:String theory
11423:
11421:
11418:
11414:
11411:
11410:
11409:
11406:
11402:
11399:
11397:
11394:
11392:
11389:
11387:
11384:
11382:
11379:
11377:
11374:
11372:
11369:
11368:
11367:
11364:
11360:
11357:
11355:
11352:
11351:
11350:
11347:
11346:
11344:
11342:
11336:
11330:
11327:
11325:
11324:Path integral
11322:
11320:
11316:
11313:
11310:
11306:
11305:Distributions
11303:
11302:
11300:
11298:
11294:
11288:
11285:
11281:
11278:
11276:
11273:
11271:
11268:
11266:
11263:
11262:
11261:
11258:
11254:
11251:
11249:
11246:
11244:
11241:
11239:
11236:
11235:
11234:
11231:
11227:
11224:
11222:
11219:
11217:
11214:
11212:
11209:
11208:
11207:
11204:
11200:
11197:
11196:
11195:
11192:
11188:
11185:
11183:
11180:
11179:
11178:
11175:
11171:
11168:
11166:
11163:
11162:
11161:
11158:
11154:
11151:
11149:
11146:
11144:
11141:
11140:
11139:
11136:
11132:
11129:
11127:
11124:
11122:
11119:
11118:
11117:
11114:
11110:
11107:
11106:
11105:
11102:
11100:
11097:
11096:
11094:
11092:
11088:
11082:
11079:
11077:
11074:
11072:
11071:Combinatorics
11069:
11067:
11064:
11062:
11059:
11058:
11056:
11054:
11050:
11040:
11037:
11035:
11032:
11030:
11027:
11025:
11022:
11020:
11017:
11015:
11012:
11010:
11007:
11005:
11002:
11000:
10997:
10996:
10994:
10992:
10988:
10982:
10979:
10977:
10974:
10972:
10969:
10967:
10964:
10960:
10957:
10956:
10955:
10952:
10950:
10947:
10945:
10944:Coding theory
10942:
10940:
10937:
10935:
10932:
10928:
10925:
10923:
10920:
10919:
10918:
10915:
10914:
10912:
10910:
10909:Computational
10906:
10902:
10895:
10890:
10888:
10883:
10881:
10876:
10875:
10872:
10860:
10857:
10855:
10852:
10850:
10847:
10845:
10842:
10840:
10837:
10836:
10834:
10830:
10824:
10821:
10819:
10818:Ocean physics
10816:
10814:
10811:
10809:
10806:
10804:
10801:
10799:
10796:
10794:
10791:
10789:
10786:
10784:
10781:
10779:
10776:
10775:
10773:
10771:
10767:
10761:
10758:
10754:
10753:Modern optics
10751:
10749:
10746:
10744:
10741:
10740:
10739:
10736:
10734:
10731:
10729:
10726:
10724:
10721:
10717:
10714:
10712:
10709:
10708:
10707:
10704:
10703:
10701:
10699:
10695:
10687:
10684:
10682:
10679:
10678:
10677:
10674:
10670:
10667:
10665:
10662:
10661:
10660:
10657:
10655:
10652:
10650:
10647:
10643:
10640:
10638:
10635:
10633:
10630:
10628:
10625:
10624:
10623:
10620:
10619:
10617:
10615:
10611:
10603:
10602:Computational
10600:
10599:
10598:
10595:
10593:
10590:
10589:
10587:
10583:
10575:
10572:
10571:
10570:
10567:
10565:
10562:
10561:
10559:
10555:
10551:
10543:
10538:
10536:
10531:
10529:
10524:
10523:
10520:
10512:
10511:0-7167-0344-0
10508:
10502:
10494:
10488:
10484:
10483:
10475:
10467:
10461:
10457:
10456:
10448:
10442:
10438:
10434:
10429:
10423:
10419:
10413:
10411:
10409:
10402:
10398:
10394:
10389:
10387:
10379:
10373:
10365:
10359:
10355:
10354:
10346:
10340:
10339:0-07-084018-0
10336:
10332:
10327:
10321:
10320:0-07-051400-3
10317:
10311:
10303:
10301:0-03-063366-4
10297:
10293:
10286:
10280:
10276:
10272:
10267:
10265:
10263:
10261:
10259:
10249:
10241:
10237:
10233:
10227:
10223:
10219:
10215:
10214:
10206:
10198:
10192:
10188:
10187:
10179:
10171:
10169:0-486-65067-7
10165:
10161:
10160:
10152:
10150:
10145:
10136:
10133:
10131:
10128:
10126:
10123:
10121:
10118:
10116:
10113:
10111:
10108:
10106:
10103:
10101:
10098:
10096:
10093:
10091:
10088:
10086:
10083:
10082:
10076:
10074:
10070:
10048:
10039:
10036:
10030:
10024:
10018:
10012:
10009:
10000:
9997:
9994:
9985:
9982:
9976:
9970:
9967:
9964:
9958:
9952:
9944:
9941:
9937:
9933:
9926:
9923:
9922:
9921:
9920:and angle θ.
9919:
9916:
9912:
9908:
9904:
9880:
9877:
9866:
9851:
9834:
9831:
9808:
9794:
9787:
9785:
9782:
9781:
9761:
9746:
9727:
9712:
9698:
9691:
9689:
9686:
9685:
9665:
9662:
9642:
9628:
9620:
9600:
9575:
9568:
9566:
9563:
9562:
9539:
9536:
9518:
9503:
9481:
9478:
9460:
9445:
9428:
9425:
9407:
9400:
9398:
9395:
9394:
9374:
9370:
9366:
9363:
9346:
9331:
9312:
9308:
9304:
9301:
9284:
9269:
9250:
9246:
9239:
9232:
9230:
9227:
9226:
9194:
9175:
9155:
9130:
9123:
9121:
9118:
9117:
9113:
9110:
9107:
9104:
9103:
9100:
9098:
9094:
9090:
9086:
9078:
9075:
9074:
9068:
9055:
9051:
9048:
9029:
9025:
9022:
9013:
9011:
9007:
8983:
8979:
8973:
8951:
8943:
8935:
8931:
8918:
8910:
8906:
8902:
8898:
8889:
8873:
8864:
8860:
8856:
8844:
8838:
8835:
8830:
8823:
8820:
8812:
8803:
8799:
8795:
8783:
8777:
8774:
8769:
8762:
8759:
8746:
8733:
8722:
8716:
8713:
8700:
8696:
8689:
8686:
8673:
8666:
8663:
8654:
8649:
8646:
8643:
8639:
8635:
8629:
8626:
8618:
8615:
8612:
8607:
8603:
8599:
8594:
8590:
8586:
8583:
8580:
8575:
8571:
8567:
8562:
8558:
8516:
8497:
8487:
8483:
8473:
8468:
8461:
8458:
8444:
8437:
8434:
8409:
8403:
8400:
8387:
8383:
8374:
8370:
8366:
8359:
8352:
8347:
8345:
8341:
8337:
8333:
8328:
8315:
8311:
8308:
8289:
8285:
8282:
8274:
8269:
8267:
8266:spinor fields
8263:
8262:tensor fields
8259:
8258:vector fields
8254:
8252:
8248:
8244:
8240:
8236:
8217:
8208:
8204:
8182:
8178:
8167:
8163:
8157:
8128:
8122:
8097:
8090:
8087:
8079:
8076:
8073:
8068:
8064:
8058:
8050:
8045:
8041:
8035:
8027:
8024:
8021:
8016:
8012:
8005:
8000:
7996:
7989:
7986:
7983:
7978:
7974:
7970:
7965:
7961:
7947:
7932:
7931:
7926:
7922:
7918:
7914:
7910:
7904:
7903:scalar fields
7901:
7894:
7884:
7882:
7872:
7868:
7860:
7856:
7852:
7828:
7824:
7820:
7813:
7801:
7797:
7790:
7785:
7778:
7766:
7761:
7747:
7746:
7745:
7728:
7722:
7717:
7705:
7701:
7695:
7690:
7687:
7684:
7680:
7674:
7671:
7666:
7663:
7659:
7650:
7646:
7637:
7628:
7623:
7607:
7606:
7605:
7588:
7579:
7575:
7571:
7564:
7560:
7554:
7550:
7543:
7538:
7531:
7528:
7521:
7516:
7512:
7504:
7503:
7502:
7500:
7495:
7485:
7482:
7477:
7475:
7471:
7452:
7435:
7426:
7417:
7403:
7391:
7388:
7384:
7375:
7374:
7373:
7371:
7367:
7363:
7344:
7327:
7318:
7315:
7309:
7297:
7280:
7271:
7268:
7262:
7247:
7246:
7245:
7243:
7239:
7220:
7210:
7201:
7193:
7185:
7174:
7171:
7156:
7148:
7145:
7138:
7137:
7136:
7134:
7130:
7120:
7113:
7109:
7105:
7095:
7088:
7084:
7080:
7070:
7063:
7059:
7055:
7049:
7039:
7037:
7033:
7029:
7025:
7020:
7018:
7014:
6995:
6992:
6989:
6983:
6980:
6962:
6948:
6941:
6940:
6939:
6938:
6918:
6914:
6911:
6882:
6873:
6869:
6866:
6860:
6857:
6849:
6838:
6835:
6832:
6825:
6824:
6823:
6821:
6799:
6778:
6775:
6772:
6765:
6764:
6763:
6761:
6716:
6704:
6703:
6702:
6685:
6681:
6678:
6672:
6669:
6651:
6645:
6642:
6629:
6625:
6617:
6616:
6615:
6613:
6585:
6581:
6574:
6570:
6569:
6568:
6566:
6562:
6543:
6540:
6532:
6528:
6524:
6519:
6515:
6504:
6503:
6502:
6500:
6496:
6492:
6488:
6484:
6480:
6475:
6473:
6469:
6465:
6461:
6454:
6435:
6418:
6409:
6406:
6390:
6373:
6364:
6361:
6342:
6341:
6340:
6338:
6334:
6308:
6305:
6297:
6284:
6280:
6273:
6261:
6255:
6252:
6244:
6233:
6230:
6224:
6221:
6213:
6202:
6191:
6188:
6180:
6167:
6163:
6156:
6144:
6138:
6135:
6127:
6116:
6113:
6107:
6104:
6096:
6085:
6074:
6071:
6063:
6050:
6046:
6039:
6027:
6021:
6018:
6010:
5999:
5996:
5990:
5987:
5979:
5968:
5957:
5954:
5946:
5933:
5929:
5922:
5910:
5904:
5901:
5893:
5882:
5879:
5873:
5870:
5862:
5851:
5839:
5838:
5837:
5835:
5831:
5827:
5823:
5819:
5815:
5811:
5807:
5803:
5799:
5795:
5791:
5787:
5783:
5779:
5772:
5769:
5768:
5762:
5760:
5756:
5752:
5748:
5744:
5741:
5736:
5710:
5706:
5678:
5671:
5664:
5645:
5641:
5638:
5635:
5632:
5626:
5623:
5608:
5597:
5590:
5586:
5578:
5574:
5569:
5565:
5562:
5552:
5545:
5544:
5543:
5541:
5540:
5520:
5516:
5513:
5507:
5504:
5489:
5478:
5471:
5467:
5459:
5455:
5450:
5446:
5432:
5431:
5430:
5428:
5424:
5417:
5413:
5409:
5405:
5400:
5388:
5384:
5368:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5338:
5334:
5318:
5299:
5288:
5284:
5272:
5267:
5260:
5257:
5248:
5241:
5238:
5231:
5226:
5216:
5209:
5206:
5193:
5188:
5184:
5180:
5169:
5161:
5154:
5151:
5144:
5139:
5132:
5129:
5122:
5116:
5111:
5101:
5094:
5091:
5084:
5075:
5067:
5063:
5059:
5056:
5053:
5037:
5034:
5026:
5019:
5016:
5004:
4992:
4989:
4985:
4980:
4974:
4971:
4964:
4960:
4956:
4945:
4942:
4934:
4930:
4921:
4907:
4903:
4899:
4896:
4892:
4876:
4869:
4866:
4858:
4847:
4841:
4829:
4826:
4823:
4820:
4812:
4808:
4804:
4800:
4799:
4794:
4791:
4787:
4785:
4781:
4767:
4760:
4757:
4746:
4740:
4737:
4733:
4728:
4725:
4722:
4718:
4715:
4704:
4698:
4695:
4694:
4689:
4686:
4682:
4678:
4674:
4670:
4666:
4662:
4658:
4654:
4650:
4646:
4639:
4635:
4628:
4624:
4617:
4616:
4615:
4593:
4580:
4574:
4565:
4550:
4546:
4535:
4532:
4529:
4519:
4518:
4517:
4515:
4511:
4507:
4503:
4499:
4495:
4476:
4469:
4461:
4452:
4446:
4443:
4440:
4434:
4428:
4425:
4420:
4417:
4407:
4406:
4405:
4403:
4399:
4395:
4372:
4363:
4359:
4350:
4336:
4332:
4323:
4314:
4306:
4302:
4293:
4279:
4275:
4266:
4255:
4251:
4247:
4245:
4224:
4215:
4199:
4190:
4174:
4165:
4149:
4140:
4138:
4126:
4113:
4110:
4107:
4101:
4095:
4092:
4089:
4075:
4074:
4073:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4011:
4007:
4001:
3977:
3966:
3955:
3952:
3949:
3946:
3943:
3940:
3936:
3930:
3927:
3917:
3908:
3897:
3891:
3870:
3869:
3868:
3866:
3865:
3860:
3856:
3852:
3848:
3844:
3843:
3837:
3835:
3832:that is, the
3816:
3807:
3804:
3794:
3783:
3769:
3759:
3749:
3735:
3734:
3733:
3731:
3730:
3724:
3722:
3718:
3714:
3710:
3691:
3682:
3672:
3661:
3647:
3646:
3645:
3643:
3642:
3621:
3610:
3591:
3590:
3589:
3571:
3564:
3556:
3547:
3544:
3538:
3535:
3530:
3527:
3517:
3516:
3515:
3513:
3509:
3502:
3498:
3491:
3472:
3455:
3446:
3443:
3427:
3410:
3401:
3398:
3379:
3378:
3377:
3375:
3371:
3355:
3329:
3326:
3311:
3300:
3297:
3282:
3274:
3268:
3265:
3257:
3246:
3239:
3238:
3237:
3220:
3211:
3207:
3203:
3198:
3194:
3190:
3185:
3181:
3174:
3170:
3161:
3154:
3151:
3139:
3130:
3127:
3119:
3112:
3109:
3097:
3088:
3080:
3073:
3070:
3058:
3048:
3044:
3021:
3012:
3000:
2999:
2998:
2996:
2995:
2990:
2986:
2982:
2978:
2974:
2968:
2958:
2956:
2952:
2949:
2930:
2919:
2908:
2905:
2902:
2899:
2896:
2893:
2889:
2883:
2873:
2867:
2849:
2848:
2847:
2845:
2841:
2837:
2833:
2829:
2825:
2822:-dimensional
2821:
2803:
2773:
2764:
2754:
2743:
2729:
2728:
2727:
2725:
2721:
2720:
2714:
2712:
2708:
2707:time integral
2703:
2701:
2697:
2690:
2687:second-order
2686:
2667:
2650:
2641:
2637:
2614:
2605:
2598:
2595:
2591:
2582:
2581:
2580:
2578:
2574:
2570:
2567:is the total
2566:
2562:
2559:is the total
2558:
2536:
2533:
2518:
2507:
2504:
2498:
2495:
2480:
2469:
2466:
2460:
2457:
2442:
2431:
2424:
2423:
2422:
2418:
2414:
2404:
2402:
2401:
2396:
2395:
2388:
2384:
2378:
2372:
2367:
2362:
2338:
2335:
2329:
2310:
2298:in the form:
2292:
2286:
2282:
2276:
2272:
2268:
2258:
2256:
2239:
2230:
2226:
2213:
2210:
2207:
2199:
2195:
2182:
2174:
2170:
2156:
2152:
2127:
2124:is the total
2123:
2118:
2105:
2088:
2079:
2075:
2052:
2043:
2036:
2033:
2029:
2024:
2009:
2008:Newton's laws
2004:
1999:
1978:
1966:
1963:
1960:
1955:
1943:
1938:
1923:
1896:
1892:
1889:
1881:
1878:
1868:
1865:
1862:
1853:
1852:
1845:
1835:
1822:
1814:
1807:
1804:
1797:
1794:
1791:
1786:
1779:
1776:
1769:
1764:
1757:
1754:
1744:
1729:
1725:
1718:
1715:
1708:
1704:
1700:
1694:
1691:
1688:
1682:
1679:
1672:
1668:
1664:
1658:
1652:
1649:
1642:
1638:
1634:
1627:
1623:
1617:
1614:
1604:
1593:
1589:
1568:
1564:
1560:
1557:
1554:
1549:
1545:
1541:
1536:
1532:
1525:
1512:
1508:
1504:
1498:
1496:
1492:
1489:= (number of
1485:
1483:
1477:
1474:
1470:
1466:
1462:
1461:configuration
1458:
1454:
1450:
1446:
1436:
1432:
1426:
1424:
1420:
1413:
1408:
1404:
1400:
1396:
1390:
1375:
1371:
1367:
1364:
1360:
1356:
1351:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1310:
1306:
1302:
1298:
1293:
1291:
1287:
1283:
1279:
1275:
1274:
1269:
1264:
1262:
1257:
1252:
1250:
1246:
1242:
1241:his third law
1238:
1233:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1194:
1192:
1181:
1178:
1163:
1160:
1152:
1149:February 2023
1141:
1138:
1134:
1131:
1127:
1124:
1120:
1117:
1113:
1110: –
1109:
1105:
1104:Find sources:
1098:
1094:
1088:
1087:
1082:This section
1080:
1076:
1071:
1070:
1061:
1058:
1050:
1047:February 2023
1040:
1036:
1032:
1026:
1023:This section
1021:
1012:
1011:
1006:
1004:
997:
996:
991:
990:
985:
980:
971:
970:
962:
958:
956:
952:
947:
945:
941:
937:
933:
927:
925:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
881:
876:
874:
870:
866:
862:
858:
853:
851:
847:
843:
839:
838:accelerations
835:
831:
826:
824:
820:
816:
812:
808:
807:
802:
798:
794:
790:
786:
774:
769:
767:
762:
760:
755:
754:
752:
751:
745:
735:
732:
727:
721:
720:
719:
718:
710:
707:
705:
702:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
591:
584:
583:
576:
572:
568:
564:
561:
560:
556:
553:
551:
548:
546:
543:
541:
538:
534:
531:
530:
529:
526:
524:
521:
519:
516:
514:
511:
510:
506:
501:
500:
492:
489:
487:
484:
480:
477:
475:
472:
471:
470:
467:
465:
462:
460:
457:
455:
450:
447:
443:
440:
439:
435:
432:
429:
425:
421:
420:
416:
413:
411:
408:
406:
403:
401:
396:
394:
391:
389:
386:
384:
381:
380:
373:
372:
361:
358:
356:
353:
351:
348:
346:
343:
341:
338:
336:
333:
332:
330:
329:
324:
321:
320:
315:
314:
308:
307:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
264:
260:
258:
255:
253:
249:
247:
244:
241:
237:
233:
231:
228:
226:
223:
221:
218:
216:
213:
209:
206:
204:
201:
200:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
175:
168:
167:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
120:
113:
112:
106:
103:
101:
98:
96:
93:
92:
90:
89:
85:
66:
63:
53:
47:
33:
32:
29:
26:
25:
21:
20:
11682: /
11678: /
11668:
11544:Optimization
11507:Superalgebra
11366:Field theory
11348:
11339:Mathematical
11317: /
11165:Chaos theory
11148:Gauge theory
11076:Graph theory
10971:Cryptography
10778:Astrophysics
10631:
10592:Experimental
10501:
10481:
10474:
10454:
10447:
10432:
10428:
10392:
10372:
10352:
10345:
10330:
10326:
10310:
10291:
10285:
10270:
10248:
10212:
10205:
10185:
10178:
10158:
10072:
10068:
9942:
9929:
9917:
9909:, θ) is the
9906:
9902:
9900:
9096:
9092:
9088:
9087:or momentum
9084:
9082:
9014:
9009:
9005:
8747:
8514:
8372:
8368:
8361:
8357:
8355:
8329:
8270:
8255:
8250:
8246:
8237:denotes the
8231:
7928:
7924:
7923:= 1, 2, ...
7920:
7916:
7912:
7905:
7899:
7897:
7877:
7870:
7863:
7858:
7854:
7850:
7848:
7743:
7603:
7497:
7480:
7478:
7473:
7469:
7467:
7369:
7365:
7361:
7359:
7241:
7237:
7235:
7132:
7125:
7118:
7111:
7107:
7100:
7093:
7086:
7082:
7075:
7068:
7061:
7057:
7053:
7051:
7021:
7016:
7010:
6936:
6934:
6819:
6817:
6759:
6757:
6700:
6611:
6583:
6579:
6577:
6564:
6560:
6558:
6498:
6494:
6490:
6486:
6482:
6478:
6476:
6471:
6467:
6462:is called a
6456:
6452:
6450:
6336:
6332:
6330:
5833:
5829:
5825:
5821:
5817:
5813:
5809:
5805:
5801:
5797:
5793:
5792:and momenta
5789:
5785:
5781:
5777:
5775:
5739:
5737:
5708:
5704:
5669:
5668:and arrival
5662:
5660:
5537:
5535:
5421:
5415:
5411:
5403:
5385:, inserting
5336:
5332:
5065:
4897:(see below).
4890:
4810:
4806:
4802:
4796:
4789:
4783:
4706:
4702:
4700:
4691:
4684:
4680:
4676:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4641:
4637:
4630:
4626:
4619:
4613:
4493:
4491:
4401:
4397:
4391:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4021:
4017:
4013:
4009:
4005:
4003:
3862:
3858:
3854:
3850:
3846:
3840:
3838:
3831:
3727:
3725:
3713:wave vectors
3708:
3706:
3639:generalized
3638:
3636:
3586:
3511:
3504:
3500:
3493:
3489:
3487:
3368:denotes the
3347:
3235:
2992:
2988:
2984:
2980:
2976:
2970:
2945:
2839:
2835:
2831:
2819:
2788:
2717:
2715:
2704:
2699:
2692:
2684:
2682:
2564:
2556:
2554:
2420:
2398:
2392:
2386:
2382:
2376:
2370:
2365:
2360:
2290:
2284:
2278:
2271:Scleronomous
2121:
2119:
2002:
1851:virtual work
1849:
1847:
1591:
1506:
1502:
1500:
1494:
1490:
1488:
1481:
1479:
1472:
1464:
1456:
1452:
1441:
1439:
1431:curvillinear
1422:
1415:
1411:
1392:
1372:
1368:
1352:
1335:
1331:
1327:
1319:
1315:
1311:
1305:well-defined
1300:
1296:
1294:
1272:
1265:
1260:
1253:
1234:
1218:"mass point"
1210:acceleration
1195:
1188:
1173:
1155:
1146:
1136:
1129:
1122:
1115:
1103:
1091:Please help
1086:verification
1083:
1053:
1044:
1031:spinning off
1024:
1000:
993:
987:
986:Please help
983:
959:
955:chaos theory
948:
928:
877:
856:
854:
849:
827:
804:
796:
792:
782:
573: /
569: /
567:displacement
565: /
426: /
388:Displacement
327:
326:
317:
311:Formulations
298:Virtual work
238: /
178:Acceleration
171:Fundamentals
11684:topics list
11618:Mathematics
11534:Game theory
11435:Topological
11401:Topological
11396:Statistical
11359:Hamiltonian
10681:Statistical
10597:Theoretical
10574:Engineering
10378:R.G. Lerner
9915:unit vector
4904:, i.e. are
4813:, that is:
4707:, that is:
4012:, and time
4008:, momentum
3729:phase space
3370:dot product
2948:topological
2394:scleronomic
2364:are called
2261:Constraints
1482:constraints
1473:generalized
1465:curvilinear
892:phase space
857:constraints
709:von Neumann
376:Core topics
11699:Categories
11590:Psychology
11554:Statistics
11354:Lagrangian
10981:Statistics
10917:Algorithms
10798:Geophysics
10788:Biophysics
10632:Analytical
10585:Approaches
10120:Kinematics
9932:continuous
9784:T-symmetry
8886:where the
8239:4-gradient
6563:= 1, 2,...
5757:motion in
5427:functional
4510:commutator
3842:phase path
3715:(given by
2275:Rheonomous
1414:, denoted
1290:invariance
1249:solid body
1119:newspapers
1035:relocating
989:improve it
924:symmetries
644:d'Alembert
624:Maupertuis
587:Scientists
469:Rigid body
143:Kinematics
11595:Sociology
11585:Chemistry
11381:Effective
11376:Conformal
11371:Classical
11243:Geometric
11216:Geometric
10748:Molecular
10649:Acoustics
10642:Continuum
10637:Celestial
10627:Newtonian
10614:Classical
10557:Divisions
10240:165699220
10040:˙
9986:˙
9940:parameter
9878:−
9867:−
9864:→
9832:−
9821:→
9762:−
9759:→
9728:−
9725:→
9655:→
9613:→
9540:θ
9531:^
9516:→
9482:θ
9473:^
9458:→
9429:θ
9420:^
9353:→
9291:→
9207:→
9168:→
9114:Momentum
9111:Position
9108:Operator
9030:∫
8980:ϕ
8974:μ
8970:∂
8963:∂
8959:∂
8952:μ
8948:∂
8944:−
8932:ϕ
8928:∂
8924:∂
8907:ϕ
8903:δ
8899:δ
8861:ϕ
8857:δ
8845:δ
8839:−
8824:˙
8821:π
8800:π
8796:δ
8784:δ
8763:˙
8760:ϕ
8723:−
8697:π
8667:˙
8664:ϕ
8640:∑
8616:…
8604:π
8591:π
8584:…
8572:ϕ
8559:ϕ
8495:∂
8484:ϕ
8480:∂
8474:≡
8462:˙
8459:ϕ
8438:˙
8435:ϕ
8428:∂
8416:∂
8384:π
8290:∫
8205:ϕ
8201:∂
8189:∂
8164:ϕ
8158:μ
8154:∂
8147:∂
8135:∂
8123:μ
8119:∂
8077:…
8065:ϕ
8055:∂
8042:ϕ
8032:∂
8025:…
8013:ϕ
8009:∇
7997:ϕ
7993:∇
7987:…
7975:ϕ
7962:ϕ
7779:¨
7681:∑
7647:α
7643:∂
7635:∂
7532:¨
7513:α
7445:ζ
7441:∂
7433:∂
7418:˙
7415:ζ
7409:∂
7401:∂
7333:∂
7325:∂
7319:−
7310:˙
7286:∂
7278:∂
7263:˙
7211:˙
7208:ζ
7198:ζ
7172:−
7166:˙
7157:⋅
6901:∂
6889:∂
6797:∂
6785:∂
6779:−
6735:∂
6723:∂
6424:∂
6416:∂
6401:˙
6379:∂
6371:∂
6365:−
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6271:∂
6267:∂
6154:∂
6150:∂
6037:∂
6033:∂
5920:∂
5916:∂
5618:˙
5570:∫
5566:δ
5553:δ
5499:˙
5451:∫
5309:˙
5261:˙
5242:˙
5210:˙
5185:λ
5155:˙
5145:λ
5133:˙
5123:λ
5095:˙
5085:λ
5020:˙
5010:∂
5002:∂
4950:→
4927:∂
4919:∂
4906:conserved
4839:∂
4835:∂
4788:describe
4584:^
4569:^
4554:ℏ
4542:→
4467:∂
4459:∂
4356:∂
4348:∂
4329:∂
4321:∂
4315:−
4299:∂
4291:∂
4272:∂
4264:∂
4252:∑
4248:≡
4230:∂
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4216:⋅
4205:∂
4197:∂
4191:−
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4155:∂
4147:∂
4102:≡
3967:⊆
3956:∈
3944:≥
3918:∈
3795:∈
3760:×
3673:∈
3605:˙
3562:∂
3554:∂
3548:−
3461:∂
3453:∂
3438:˙
3416:∂
3408:∂
3402:−
3393:˙
3356:⋅
3321:˙
3298:−
3292:˙
3283:⋅
3204:⋯
3155:˙
3145:∂
3137:∂
3131:⋯
3113:˙
3103:∂
3095:∂
3074:˙
3064:∂
3056:∂
3036:˙
3027:∂
3019:∂
2951:manifolds
2920:⊆
2909:∈
2897:≥
2874:∈
2755:∈
2656:∂
2648:∂
2629:˙
2620:∂
2612:∂
2528:˙
2505:−
2490:˙
2452:˙
2400:rheonomic
2368:. Vector
2294:and time
2223:∂
2219:∂
2211:…
2192:∂
2188:∂
2167:∂
2163:∂
2142:∂
2138:∂
2094:∂
2086:∂
2080:−
2067:˙
2058:∂
2050:∂
1964:…
1882:δ
1879:⋅
1863:δ
1808:˙
1795:…
1780:˙
1758:˙
1739:˙
1730:≡
1692:…
1558:…
1469:dimension
1455:= 1, 2...
1447:for each
1245:rotations
995:talk page
823:variation
689:Liouville
571:frequency
491:Vibration
208:potential
133:Continuum
128:Celestial
105:Textbooks
11670:Category
11319:analysis
11238:Exterior
11211:Exterior
11091:Analysis
11053:Discrete
10927:analysis
10105:Dynamics
10079:See also
8241:and the
7919:) where
7133:Routhian
6559:for all
6466:of the "
5755:geodesic
4719:′
2983:) with (
2953:and the
1996:are the
1586:and the
1407:position
1309:function
1261:a priori
1222:particle
1206:velocity
873:friction
744:Category
669:Hamilton
654:Lagrange
649:Clairaut
614:Horrocks
575:velocity
545:Pendulum
533:reactive
505:Rotation
474:dynamics
424:Inertial
410:Friction
293:Velocity
268:Momentum
148:Kinetics
138:Dynamics
116:Branches
100:Timeline
11680:outline
11611:Related
11580:Biology
11430:Bosonic
11391:Quantum
11341:physics
11307: (
11039:Solvers
10832:Related
10716:General
10711:Special
10569:Applied
8375:) are:
7124:, ...,
4798:partial
2358:, then
1133:scholar
842:momenta
704:Koopman
664:Poisson
659:Laplace
604:Huygens
599:Galileo
444: (
383:Damping
236:Inertia
230:Impulse
203:kinetic
153:Statics
123:Applied
95:History
11253:Vector
11248:Tensor
11226:Vector
11221:Tensor
10922:design
10743:Atomic
10698:Modern
10548:Major
10509:
10489:
10462:
10439:
10420:
10399:
10360:
10337:
10318:
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10277:
10238:
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10193:
10166:
9901:where
9688:Parity
8342:, and
8264:, and
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7927:, the
7744:where
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6818:where
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5423:Action
5331:where
5052:cyclic
4679:) and
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3503:) and
3348:where
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1198:vector
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846:forces
834:vector
817:. The
806:scalar
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694:Appell
679:Cauchy
674:Jacobi
619:Halley
609:Newton
594:Kepler
446:linear
442:Motion
288:Torque
263:Moment
198:Energy
188:Couple
11386:Gauge
7128:N − s
4693:total
3721:waves
1511:tuple
1288:, an
1247:of a
1237:fluid
1226:Earth
1202:force
1140:JSTOR
1126:books
795:, or
699:Gibbs
684:Routh
639:Euler
278:Speed
273:Space
215:Force
10669:Wave
10564:Pure
10507:ISBN
10487:ISBN
10460:ISBN
10437:ISBN
10418:ISBN
10397:ISBN
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10335:ISBN
10316:ISBN
10296:ISBN
10275:ISBN
10236:OCLC
10226:ISBN
10191:ISBN
10164:ISBN
7360:and
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4809:and
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283:Time
246:Mass
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10218:doi
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8232:∂
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7900:N
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7767:=
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7688:=
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7667:=
7664:S
7660:,
7651:r
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7629:=
7624:r
7618:Q
7589:,
7580:2
7576:t
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7544:=
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7522:=
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7470:N
7453:.
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7427:=
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7392:t
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7316:=
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6956:q
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6836:=
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6776:=
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6717:=
6713:p
6686:,
6682:C
6679:+
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6657:S
6652:=
6649:)
6646:t
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6639:q
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6630:2
6626:G
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6596:S
6580:K
6565:N
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6541:=
6538:}
6533:i
6529:P
6525:,
6520:i
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6499:P
6495:p
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6407:=
6398:Q
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6362:=
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6312:)
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6225:t
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6000:H
5997:=
5994:)
5991:t
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5984:P
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5976:Q
5972:(
5969:K
5961:)
5958:t
5955:,
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5880:=
5877:)
5874:t
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5867:P
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5859:Q
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5639:=
5636:t
5633:d
5630:)
5627:t
5624:,
5615:q
5609:,
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5601:(
5598:L
5591:2
5587:t
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5575:t
5563:=
5558:S
5521:.
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5508:t
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5496:q
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5482:(
5479:L
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5447:=
5442:S
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5412:S
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5369:.
5365:E
5362:=
5359:V
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5353:T
5350:=
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5319:,
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4889:(
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4673:q
4669:t
4667:(
4665:p
4661:t
4659:(
4657:q
4653:t
4649:t
4647:(
4644:i
4642:p
4638:t
4636:(
4633:i
4627:t
4625:(
4622:i
4620:q
4594:.
4590:]
4581:B
4575:,
4566:A
4560:[
4551:i
4547:1
4539:}
4536:B
4533:,
4530:A
4527:{
4494:A
4477:.
4470:t
4462:A
4453:+
4450:}
4447:H
4444:,
4441:A
4438:{
4435:=
4429:t
4426:d
4421:A
4418:d
4402:A
4398:A
4373:,
4364:k
4360:q
4351:B
4337:k
4333:p
4324:A
4307:k
4303:p
4294:B
4280:k
4276:q
4267:A
4256:k
4234:q
4225:B
4209:p
4200:A
4184:p
4175:B
4159:q
4150:A
4141:=
4131:p
4127:,
4123:q
4118:}
4114:B
4111:,
4108:A
4105:{
4099:}
4096:B
4093:,
4090:A
4087:{
4066:t
4062:p
4058:q
4056:(
4054:B
4050:t
4046:p
4042:q
4040:(
4038:A
4034:t
4030:p
4026:q
4024:(
4022:A
4018:A
4014:t
4010:p
4006:q
3978:,
3972:P
3964:}
3960:R
3953:t
3950:,
3947:0
3941:t
3937::
3931:N
3928:2
3923:R
3915:)
3912:)
3909:t
3906:(
3902:p
3898:,
3895:)
3892:t
3889:(
3885:q
3881:(
3878:{
3859:t
3857:(
3855:p
3851:t
3849:(
3847:q
3817:,
3813:}
3808:N
3805:2
3800:R
3792:)
3788:p
3784:,
3780:q
3776:(
3773:{
3770:=
3765:M
3755:C
3750:=
3745:P
3709:k
3692:.
3688:}
3683:N
3678:R
3669:p
3665:{
3662:=
3657:M
3622:.
3616:Q
3611:=
3602:p
3572:,
3565:t
3557:L
3545:=
3539:t
3536:d
3531:H
3528:d
3512:t
3510:(
3507:i
3505:p
3501:t
3499:(
3496:i
3494:q
3490:N
3473:,
3465:p
3456:H
3447:+
3444:=
3435:q
3428:,
3420:q
3411:H
3399:=
3390:p
3333:)
3330:t
3327:,
3318:q
3312:,
3308:q
3304:(
3301:L
3289:q
3279:p
3275:=
3272:)
3269:t
3266:,
3262:p
3258:,
3254:q
3250:(
3247:H
3221:,
3217:)
3212:N
3208:p
3199:2
3195:p
3191:,
3186:1
3182:p
3178:(
3175:=
3171:)
3162:N
3152:q
3140:L
3128:,
3120:2
3110:q
3098:L
3089:,
3081:1
3071:q
3059:L
3049:(
3045:=
3033:q
3022:L
3013:=
3009:p
2989:p
2985:q
2977:q
2931:,
2925:C
2917:}
2913:R
2906:t
2903:,
2900:0
2894:t
2890::
2884:N
2879:R
2871:)
2868:t
2865:(
2861:q
2857:{
2840:t
2838:(
2836:q
2820:N
2804:N
2799:R
2774:,
2770:}
2765:N
2760:R
2751:q
2747:{
2744:=
2739:C
2700:t
2698:(
2695:i
2693:q
2685:N
2668:,
2660:q
2651:L
2642:=
2638:)
2626:q
2615:L
2606:(
2599:t
2596:d
2592:d
2565:V
2557:T
2540:)
2537:t
2534:,
2525:q
2519:,
2515:q
2511:(
2508:V
2502:)
2499:t
2496:,
2487:q
2481:,
2477:q
2473:(
2470:T
2467:=
2464:)
2461:t
2458:,
2449:q
2443:,
2439:q
2435:(
2432:L
2389:)
2387:t
2385:(
2383:q
2377:t
2371:r
2361:q
2356:t
2342:)
2339:t
2336:,
2333:)
2330:t
2327:(
2323:q
2319:(
2315:r
2311:=
2307:r
2296:t
2291:q
2285:r
2240:)
2231:N
2227:q
2214:,
2208:,
2200:2
2196:q
2183:,
2175:1
2171:q
2157:(
2153:=
2146:q
2122:T
2106:,
2098:q
2089:T
2076:)
2064:q
2053:T
2044:(
2037:t
2034:d
2030:d
2025:=
2020:Q
2003:q
1984:)
1979:N
1973:Q
1967:,
1961:,
1956:2
1950:Q
1944:,
1939:1
1933:Q
1927:(
1924:=
1919:Q
1897:,
1893:0
1890:=
1886:q
1874:Q
1869:=
1866:W
1823:.
1820:)
1815:N
1805:q
1798:,
1792:,
1787:2
1777:q
1770:,
1765:1
1755:q
1748:(
1745:=
1736:q
1726:)
1719:t
1716:d
1709:N
1705:q
1701:d
1695:,
1689:,
1683:t
1680:d
1673:2
1669:q
1665:d
1659:,
1653:t
1650:d
1643:1
1639:q
1635:d
1628:(
1624:=
1618:t
1615:d
1609:q
1605:d
1574:)
1569:N
1565:q
1561:,
1555:,
1550:2
1546:q
1542:,
1537:1
1533:q
1529:(
1526:=
1522:q
1509:-
1507:N
1503:N
1497:)
1484:)
1457:N
1453:i
1444:i
1442:q
1423:i
1421:(
1418:i
1416:q
1336:t
1332:t
1328:t
1322:(
1320:t
1316:t
1314:(
1312:f
1301:t
1297:t
1180:)
1174:(
1162:)
1156:(
1151:)
1147:(
1137:·
1130:·
1123:·
1116:·
1089:.
1060:)
1054:(
1049:)
1045:(
1041:.
1027:.
1005:)
1001:(
772:e
765:t
758:v
448:)
67:t
64:d
58:p
54:d
48:=
43:F
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