744:
3323:
2038:
757:
7626:
3075:
1759:
873:
1421:
4775:
1668:
7202:
5873:
5636:
3023:
7153:
6368:
3318:{\displaystyle X_{H}={\frac {\partial H}{\partial p_{\mu }}}{\frac {\partial }{\partial q^{\mu }}}-{\frac {\partial H}{\partial q^{\mu }}}{\frac {\partial }{\partial p_{\mu }}}={\frac {dq^{\mu }}{dt}}{\frac {\partial }{\partial q^{\mu }}}+{\frac {dp^{\mu }}{dt}}{\frac {\partial }{\partial p_{\mu }}}={\frac {d}{dt}}}
852:. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general setting for conjugate position and momentum coordinates is available in the mathematical setting of
5490:
6373:
Unlike the equations of motion for the simple harmonic oscillator, these modified equations do not take the form of
Hamilton's equations, and therefore we do not expect Liouville's theorem to hold. Instead, as depicted in the animation in this section, a generic phase space volume will shrink as it
7162:
As for how the phase-space volume evolves in time, we will still have the constant rotation as in the undamped case. However, the damping will introduce a steady decrease in the radii of each ellipse. Again we can solve for the trajectories explicitly using
Hamilton's equations, taking care to use
7771:
can be interpreted as
Liouville's theorem, which reduces the task of solving the Vlasov equation to that of single particle motion. By using Liouville's theorem in this way with energy or magnetic moment conservation, for example, one can determine unknown fields using known particle distribution
2033:{\displaystyle \rho \sum _{i=1}^{n}\left({\frac {\partial {\dot {q}}_{i}}{\partial q_{i}}}+{\frac {\partial {\dot {p}}_{i}}{\partial p_{i}}}\right)=\rho \sum _{i=1}^{n}\left({\frac {\partial ^{2}H}{\partial q_{i}\,\partial p_{i}}}-{\frac {\partial ^{2}H}{\partial p_{i}\partial q_{i}}}\right)=0,}
1231:
5644:
The question remains of how the phase-space volume actually evolves in time. Above we have shown that the total volume is conserved, but said nothing about what it looks like. For a single particle we can see that its trajectory in phase space is given by the ellipse of constant
907:
was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics. It is referred to as the
Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838. Consider a
4544:
6055:
4325:
7158:
We have found that the infinitesimal phase-space volume is no longer constant, and thus the phase-space density is not conserved. As can be seen from the equation as time increases, we expect our phase-space volume to decrease to zero as friction affects the system.
1489:
7621:{\displaystyle {\begin{aligned}q_{i}(t)&=e^{-\alpha t}\left&&\omega _{1}\equiv {\sqrt {\omega ^{2}-\alpha ^{2}}},\\p_{i}(t)&=e^{-\alpha t}\left&&B_{i}\equiv {\frac {1}{\omega _{1}}}\left({\frac {P_{i}}{m}}+2\alpha Q_{i}\right),\end{aligned}}}
5217:
6878:
3807:
5671:
3560:. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by
2894:
5501:
6213:
2622:
2546:
6912:
4066:
3749:
5952:-th particle. For a system of multiple particles, each one will have a phase-space trajectory that traces out an ellipse corresponding to the particle's energy. The frequency at which the ellipse is traced is given by the
5315:
3638:
833:—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical
4114:
1416:{\displaystyle {\frac {d\rho }{dt}}={\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{n}\left({\frac {\partial \rho }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \rho }{\partial p_{i}}}{\dot {p}}_{i}\right)=0.}
2340:
1110:
1747:
6580:
5036:
4770:{\displaystyle {\begin{cases}dq_{i}'=dq_{i}+{\frac {\partial {\dot {q_{i}}}}{\partial q_{i}}}dq_{i}\delta t,\\dp_{i}'=dp_{i}+{\frac {\partial {\dot {p_{i}}}}{\partial p_{i}}}dp_{i}\delta t.\end{cases}}}
98:
122:
5304:
3455:
2175:
2118:
1663:{\displaystyle {\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{n}\left({\frac {\partial (\rho {\dot {q}}_{i})}{\partial q_{i}}}+{\frac {\partial (\rho {\dot {p}}_{i})}{\partial p_{i}}}\right)=0.}
1158:
7207:
6596:
6218:
5676:
7194:
3534:
2833:
6024:
7859:
J. W. Gibbs, "On the
Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics." Proceedings of the American Association for the Advancement of Science,
5090:
5868:{\displaystyle {\begin{aligned}q_{i}(t)&=Q_{i}\cos {\omega t}+{\frac {P_{i}}{m\omega }}\sin {\omega t},\\p_{i}(t)&=P_{i}\cos {\omega t}-m\omega Q_{i}\sin {\omega t},\end{aligned}}}
6100:
4180:
6150:-dimensional isotropic harmonic potential, the Hamiltonian for which is given in the previous example. This time, we add the condition that each particle experiences a frictional force
3969:
3940:
4808:
2263:
6591:
4488:
4444:
4536:
3018:{\displaystyle X_{f}={\frac {\partial f}{\partial p_{\mu }}}{\frac {\partial }{\partial q^{\mu }}}-{\frac {\partial f}{\partial q^{\mu }}}{\frac {\partial }{\partial p_{\mu }}}.}
2720:
5631:{\displaystyle \rho (\Gamma ',t+\delta t)={\frac {\mathrm {d} {\mathcal {N}}}{\mathrm {d} \Gamma '}}={\frac {\mathrm {d} {\mathcal {N}}}{\mathrm {d} \Gamma }}=\rho (\Gamma ,t),}
2664:. More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes a corollary.
6363:{\displaystyle {\begin{aligned}{\dot {q_{i}}}&={\frac {\partial H}{\partial p_{i}}},\\{\dot {p_{i}}}&=-{\frac {\partial H}{\partial q_{i}}}-\gamma p_{i}.\end{aligned}}}
4149:
4400:
4364:
6181:
1203:
1008:
3840:
3392:
2772:
2557:
7787:
1043:
2213:
6904:
4831:
4511:
4172:
3067:
6201:
6044:
5970:
3887:
3478:
2886:
7683:
7656:
5930:
5903:
3360:
1473:
970:
940:
2177:
have been evaluated using
Hamilton's equations of motion. That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the
7885:
7148:{\displaystyle \mathrm {d} \Gamma '=\displaystyle \prod _{i=1}^{N}d^{3}q_{i}'d^{3}p_{i}'=\left^{3N}\prod _{i=1}^{N}d^{3}q_{i}d^{3}p_{i}=\mathrm {d} \Gamma \left.}
6377:
To see this violation of
Liouville's theorem explicitly, we can follow a very similar procedure to the undamped harmonic oscillator case, and we arrive again at
7703:
6148:
6128:
6110:
apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of
5950:
5663:
5078:
5059:
3911:
3772:
2740:
2359:
2061:
1445:
1223:
788:
3977:
3668:
377:
3793:, and thus violations of Liouville's theorem incompressibility. This, then, leads to concomitant difficulties in defining meaningful quantum trajectories.
861:
496:
469:
7950:
Harald J.W. Müller-Kirsten, Introduction to
Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012).
5485:{\displaystyle \mathrm {d} \Gamma '=\displaystyle \prod _{i=1}^{N}d^{3}q_{i}'d^{3}p_{i}'=\prod _{i=1}^{N}d^{3}q_{i}d^{3}p_{i}=\mathrm {d} \Gamma .}
8336:
8039:
Oliva, Maxime; Kakofengitis, Dimitris; Steuernagel, Ole (2018). "Anharmonic quantum mechanical systems do not feature phase space trajectories".
7932:
31:
3570:
8331:
8321:
4074:
3774:
is an observable. Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent.
743:
8249:, edited by G. Paschmann and P. W. Daly, no. SR-001 in ISSI Scientific Reports, chap. 7, pp. 159–163, ESA Publ. Div., Noordwijk, Netherlands.
8104:
7969:
7715:
2288:
6058:
The evolution of phase-space volume for the damped harmonic oscillator. The same values of parameters are used as in the SHO case, with
849:
451:
5222:
By using
Hamilton's equations with the above Hamiltonian we find that the term in parentheses above is identically zero, thus yielding
5972:
in the
Hamiltonian, independent of any differences in energy. As a result, a region of phase space will simply rotate about the point
1048:
7705:-th particle. As the system evolves the total phase-space volume will spiral in to the origin. This can be seen in the figure above.
2640:, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In
1676:
8287:
8150:
781:
892:, Liouville's equation describes the flow of the whole distribution. The motion is analogous to a dye in an incompressible fluid.
899:
117:
54:
8326:
112:
3400:
2123:
2066:
372:
1115:
5212:{\displaystyle H=\displaystyle \sum _{i=1}^{3N}\left({\frac {1}{2m}}p_{i}^{2}+{\frac {m\omega ^{2}}{2}}q_{i}^{2}\right).}
864:. There are extensions of Liouville's theorem to cover these various generalized settings, including stochastic systems.
367:
7782:
7726:
7166:
3486:
2783:
774:
761:
522:
445:
6383:
5975:
4839:
210:
441:
242:
4320:{\displaystyle {\begin{cases}q_{i}'=q_{i}+{\dot {q_{i}}}\delta t,\\p_{i}'=p_{i}+{\dot {p_{i}}}\delta t,\end{cases}}}
4151:
is constant along the trajectories of the system. If we allow our particles to evolve by an infinitesimal time step
7891:
5081:
661:
550:
476:
336:
269:
5228:
7714:
The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is a fundamental equation of
6061:
415:
8304:
8192:
7730:
3945:
3916:
2842:
of the symplectic 2-form, and is just another representation of the measure on the phase space described above.
2850:
691:
535:
6873:{\displaystyle {\begin{aligned}dq_{i}'dp_{i}'&=dq_{i}dp_{i}\left,\\&=dq_{i}dp_{i}\left.\end{aligned}}}
2218:
8245:
Schwartz, S. J., Daly, P. W., and Fazakerley, A. N., 1998, Multi-Spacecraft Analysis of Plasma Kinetics, in
7989:
7746:
3790:
3778:
2645:
1427:
681:
641:
405:
6204:
4783:
4449:
4405:
3659:
3553:
889:
686:
503:
7868:
4516:
2679:
7757:
7753:
7734:
5080:-dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a
2649:
2641:
2178:
913:
877:
818:
814:
696:
671:
357:
175:
7807:
Harald J. W. Müller-Kirsten, Basics of Statistical Physics, 2nd ed., World Scientific (Singapore, 2013)
5041:
So far, we have yet to make any specifications about our system. Let us now specialize to the case of
4119:
8058:
7881:
7826:
7742:
3557:
904:
834:
716:
711:
676:
584:
580:
572:
562:
352:
345:
101:
4553:
4369:
4333:
4189:
2617:{\displaystyle {\frac {\partial \rho }{\partial t}}+{\mathrm {i} {\widehat {\mathbf {L} }}}\rho =0.}
7761:
7722:
6153:
2846:
2673:
1480:
853:
841:
491:
432:
410:
155:
150:
145:
45:
1167:
975:
888:(red curve, lower figure). Whereas the motion of an individual member of the ensemble is given by
860:, where the total number of particles may change over time, or where energy may be transferred to
8277:
8204:
A rigorous proof based on how the Jacobian volume element transforms under Hamiltonian mechanics.
8074:
8048:
3813:
3365:
2745:
1449:
1430:
for the system. This equation demonstrates the conservation of density in phase space (which was
909:
621:
362:
237:
205:
165:
1013:
5495:
Thus we have ultimately found that the infinitesimal phase-space volume is unchanged, yielding
3810:
The time evolution of phase space for the simple harmonic oscillator (SHO). Here we have taken
3480:
itself is preserved, not only its top exterior power. That is, Liouville's Theorem also gives
2184:
8283:
8273:
8146:
8136:
8121:
8100:
8010:
7965:
7842:
6886:
4813:
4493:
4154:
3651:
3545:
3031:
2637:
1750:
943:
857:
631:
588:
545:
540:
481:
257:
247:
140:
8140:
8026:
6186:
6029:
5955:
3845:
3463:
8066:
7834:
2856:
2661:
2657:
810:
726:
706:
651:
646:
592:
567:
422:
280:
225:
200:
8214:
7906:
7661:
7634:
5908:
5881:
3338:
1458:
948:
918:
848:
are absent or can be ignored. The general mathematical formulation for such systems is the
7768:
3786:
2541:{\displaystyle \mathrm {i} {\widehat {\mathbf {L} }}=\sum _{i=1}^{n}\left=-\{H,\bullet \}}
2279:
721:
666:
616:
611:
530:
8166:
8062:
7830:
30:
This article is about Liouville's theorem in Hamiltonian mechanics. For other uses, see
8178:
Nearly identical to proof in this Knowledge (XXG) article. Assumes (without proof) the
7688:
6133:
6113:
6054:
5935:
5648:
5063:
5044:
4061:{\displaystyle \mathrm {d} \Gamma =\displaystyle \prod _{i=1}^{N}d^{3}p_{i}d^{3}q_{i}.}
3896:
3757:
3744:{\displaystyle {\frac {d}{dt}}\langle A\rangle =-{\frac {1}{i\hbar }}\langle \rangle ,}
3644:
3549:
3332:
3328:
where we utilized Hamilton's equations of motion and the definition of the chain rule.
2839:
2725:
2633:
2046:
1208:
885:
748:
656:
557:
274:
220:
6883:
Calculating our new infinitesimal phase space volume, and keeping only first order in
2215:, is zero follows from the equation of continuity by noting that the 'velocity field'
8315:
8092:
3782:
1431:
636:
463:
8078:
7931:
Ehrendorfer, Martin. "The Liouville Equation: Background - Historical Background".
3806:
701:
626:
315:
195:
872:
1753:. Notice that the difference between this and Liouville's equation are the terms
8070:
881:
824:
7997:
Proves Liouville's theorem using the language of modern differential geometry.
3655:
3561:
2265:
in phase space has zero divergence (which follows from Hamilton's relations).
828:
486:
160:
7846:
7738:
903:. Although the equation is usually referred to as the "Liouville equation",
508:
17:
3633:{\displaystyle {\frac {\partial \rho }{\partial t}}={\frac {1}{i\hbar }},}
1439:
The distribution function is constant along any trajectory in phase space.
5665:. Explicitly, one can solve Hamilton's equations for the system and find
4109:{\displaystyle {\frac {\mathrm {d} {\mathcal {N}}}{\mathrm {d} \Gamma }}}
3913:-particle system in three dimensions, and focus on only the evolution of
884:(top). Each system consists of one massive particle in a one-dimensional
845:
427:
310:
285:
802:
400:
253:
170:
7838:
6203:
is a positive constant dictating the amount of friction. As this is a
3028:
Specifically, when the generating function is the Hamiltonian itself,
1112:
that the system will be found in the infinitesimal phase space volume
2775:
1426:
Time derivatives are denoted by dots, and are evaluated according to
459:
305:
215:
7772:
functions, or vice versa. This method is known as Liouville mapping.
2335:{\displaystyle {\frac {\partial \rho }{\partial t}}=\{\,H,\rho \,\}}
8053:
7729:
can be derived. It is also the key component of the derivation of
295:
290:
232:
4810:, we need the product of the above quantities. To first order in
3789:
in the phase-space analog of the von Neumann equation results in
1105:{\displaystyle \rho (p,q)\;\mathrm {d} ^{n}q\,\mathrm {d} ^{n}p}
300:
263:
5309:
From this we can find the infinitesimal volume of phase space:
2660:). The theorem says this smooth measure is invariant under the
1742:{\displaystyle (\rho ,\rho {\dot {q}}_{i},\rho {\dot {p}}_{i})}
5641:
demonstrating that Liouville's theorem holds for this system.
27:
Key result in Hamiltonian mechanics and statistical mechanics
8305:"Phase space distribution functions and Liouville's theorem"
8193:"Preservation of phase space volume and Liouville's theorem"
7964:(2 ed.). Taylor & Francis Group. pp. 201–204.
7817:
Kubo, Ryogo (1963-02-01). "Stochastic Liouville Equations".
5586:
5550:
4174:, we see that each particle phase space location changes as
4088:
3956:
3927:
3493:
3407:
7907:"Sur la Theorie de la Variation des constantes arbitraires"
4763:
4313:
896:
The Liouville equation describes the time evolution of the
1434:'s name for the theorem). Liouville's theorem states that
3556:
yields a quantum-mechanical version of this theorem, the
1455:. This proof is based on the fact that the evolution of
93:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
7721:
The Liouville equation is integral to the proof of the
6585:
Plugging in our modified Hamilton's equations, we find
3335:
of the volume form is zero along the flow generated by
3331:
In this formalism, Liouville's Theorem states that the
3450:{\displaystyle {\mathcal {L}}_{X_{H}}(\omega ^{n})=0.}
2676:. For a given system, we can consider the phase space
2672:
We can also formulate Liouville's Theorem in terms of
2170:{\displaystyle \partial {\dot {p}}_{i}/\partial p_{i}}
2113:{\displaystyle \partial {\dot {q}}_{i}/\partial q_{i}}
8041:
Physica A: Statistical Mechanics and Its Applications
7691:
7664:
7637:
7205:
7169:
6932:
6915:
6889:
6594:
6386:
6216:
6189:
6156:
6136:
6116:
6064:
6032:
5978:
5958:
5938:
5911:
5884:
5674:
5651:
5504:
5335:
5318:
5231:
5100:
5093:
5066:
5047:
4842:
4816:
4786:
4547:
4519:
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4452:
4408:
4372:
4336:
4183:
4157:
4122:
4077:
3992:
3980:
3948:
3919:
3899:
3848:
3816:
3760:
3671:
3573:
3489:
3466:
3403:
3368:
3341:
3078:
3034:
2897:
2859:
2786:
2748:
2728:
2682:
2560:
2362:
2291:
2221:
2187:
2126:
2069:
2049:
1762:
1679:
1492:
1461:
1234:
1211:
1170:
1118:
1051:
1016:
978:
951:
921:
57:
7934:
The Liouville Equation in Atmospheric Predictability
4490:
respectively, and we have only kept terms linear in
2278:
The theorem above is often restated in terms of the
1153:{\displaystyle \mathrm {d} ^{n}q\,\mathrm {d} ^{n}p}
7697:
7677:
7650:
7620:
7189:{\displaystyle \alpha \equiv {\frac {\gamma }{2}}}
7188:
7147:
6898:
6872:
6574:
6362:
6195:
6175:
6142:
6122:
6094:
6038:
6018:
5964:
5944:
5924:
5897:
5867:
5657:
5630:
5484:
5298:
5211:
5072:
5053:
5030:
4825:
4802:
4769:
4530:
4505:
4482:
4438:
4394:
4358:
4319:
4166:
4143:
4108:
4060:
3971:particles occupy an infinitesimal volume given by
3963:
3934:
3905:
3881:
3834:
3766:
3743:
3632:
3529:{\displaystyle {\mathcal {L}}_{X_{H}}(\omega )=0.}
3528:
3472:
3449:
3386:
3354:
3317:
3061:
3017:
2880:
2828:{\displaystyle \omega =dp_{\mu }\wedge dq^{\mu }.}
2827:
2766:
2734:
2714:
2616:
2540:
2334:
2257:
2207:
2169:
2112:
2055:
2032:
1741:
1662:
1467:
1415:
1217:
1197:
1152:
1104:
1037:
1002:
964:
934:
92:
8099:. University of Cambridge Press. pp. 59–60.
7865:The Scientific Papers of J. Willard Gibbs, Vol II
7788:Reversible reference system propagation algorithm
6575:{\displaystyle dq_{i}'dp_{i}'=dq_{i}dp_{i}\left.}
6106:To see an example where Liouville's theorem does
6019:{\displaystyle (\mathbf {q} ,\mathbf {p} )=(0,0)}
5031:{\displaystyle dq_{i}'dp_{i}'=dq_{i}dp_{i}\left.}
4780:To find the new infinitesimal phase-space volume
856:. Liouville's theorem ignores the possibility of
8233:-dimensional divergence theorem (without proof).
7685:denote the initial position and momentum of the
5932:denote the initial position and momentum of the
4513:. Extending this to our infinitesimal hypercube
7887:Elementary Principles in Statistical Mechanics
5084:. The Hamiltonian for this system is given by
2063:is the Hamiltonian, and where the derivatives
782:
8:
7914:Journal de mathématiques pures et appliquées
6207:, we need to extend Hamilton's equations as
5299:{\displaystyle dq_{i}'dp_{i}'=dq_{i}dp_{i}.}
4116:to remain the same throughout time, so that
3735:
3717:
3693:
3687:
2648:that comes naturally equipped with a smooth
2535:
2523:
2329:
2315:
827:distribution function is constant along the
6095:{\displaystyle \gamma =0.5\ (\alpha =0.25)}
6046:. This can be seen in the animation above.
3564:. In this case, the resulting equation is
2838:The volume form of our manifold is the top
844:, that is, systems in which the effects of
8247:Analysis Methods for Multi-Spacecraft Data
7983:
7981:
3964:{\displaystyle \mathrm {d} {\mathcal {N}}}
3935:{\displaystyle \mathrm {d} {\mathcal {N}}}
1070:
789:
775:
36:
8052:
7690:
7669:
7663:
7642:
7636:
7600:
7576:
7570:
7557:
7548:
7539:
7517:
7512:
7497:
7478:
7468:
7445:
7440:
7428:
7407:
7381:
7362:
7349:
7343:
7334:
7312:
7307:
7295:
7278:
7273:
7261:
7240:
7214:
7206:
7204:
7176:
7168:
7102:
7093:
7083:
7073:
7063:
7053:
7042:
7029:
6991:
6981:
6968:
6958:
6948:
6937:
6916:
6914:
6888:
6832:
6819:
6768:
6755:
6737:
6730:
6718:
6705:
6687:
6680:
6658:
6645:
6622:
6606:
6595:
6593:
6544:
6523:
6517:
6516:
6510:
6498:
6477:
6471:
6470:
6464:
6442:
6429:
6410:
6394:
6385:
6374:evolves under these equations of motion.
6347:
6328:
6310:
6288:
6282:
6281:
6265:
6247:
6228:
6222:
6221:
6217:
6215:
6188:
6167:
6155:
6135:
6115:
6063:
6031:
5990:
5982:
5977:
5957:
5937:
5916:
5910:
5889:
5883:
5850:
5838:
5817:
5805:
5779:
5760:
5738:
5732:
5721:
5709:
5683:
5675:
5673:
5650:
5593:
5585:
5584:
5579:
5576:
5557:
5549:
5548:
5543:
5540:
5503:
5470:
5461:
5451:
5441:
5431:
5421:
5410:
5394:
5384:
5371:
5361:
5351:
5340:
5319:
5317:
5287:
5274:
5255:
5239:
5230:
5194:
5189:
5173:
5163:
5154:
5149:
5130:
5116:
5105:
5092:
5065:
5046:
5000:
4979:
4973:
4972:
4966:
4954:
4933:
4927:
4926:
4920:
4898:
4885:
4866:
4850:
4841:
4815:
4787:
4785:
4745:
4729:
4708:
4702:
4701:
4695:
4686:
4667:
4641:
4625:
4604:
4598:
4597:
4591:
4582:
4563:
4548:
4546:
4520:
4518:
4495:
4463:
4453:
4451:
4419:
4409:
4407:
4380:
4374:
4373:
4371:
4344:
4338:
4337:
4335:
4289:
4283:
4282:
4273:
4257:
4228:
4222:
4221:
4212:
4196:
4184:
4182:
4156:
4121:
4095:
4087:
4086:
4081:
4078:
4076:
4048:
4038:
4028:
4018:
4008:
3997:
3981:
3979:
3955:
3954:
3949:
3947:
3926:
3925:
3920:
3918:
3898:
3847:
3815:
3759:
3702:
3672:
3670:
3658:, the corresponding equation is given by
3597:
3574:
3572:
3503:
3498:
3492:
3491:
3488:
3465:
3432:
3417:
3412:
3406:
3405:
3402:
3367:
3346:
3340:
3300:
3288:
3275:
3258:
3248:
3236:
3223:
3206:
3196:
3184:
3171:
3162:
3144:
3132:
3119:
3110:
3092:
3083:
3077:
3033:
3003:
2990:
2981:
2963:
2951:
2938:
2929:
2911:
2902:
2896:
2858:
2816:
2800:
2785:
2747:
2727:
2703:
2690:
2681:
2593:
2591:
2590:
2585:
2584:
2561:
2559:
2503:
2490:
2481:
2463:
2451:
2438:
2429:
2411:
2400:
2389:
2371:
2369:
2368:
2363:
2361:
2328:
2318:
2292:
2290:
2241:
2240:
2226:
2225:
2220:
2194:
2186:
2161:
2149:
2143:
2132:
2131:
2125:
2104:
2092:
2086:
2075:
2074:
2068:
2048:
2007:
1994:
1976:
1969:
1957:
1949:
1943:
1925:
1918:
1907:
1896:
1872:
1857:
1846:
1845:
1838:
1826:
1811:
1800:
1799:
1792:
1781:
1770:
1761:
1730:
1719:
1718:
1705:
1694:
1693:
1678:
1640:
1622:
1611:
1610:
1597:
1585:
1567:
1556:
1555:
1542:
1531:
1520:
1493:
1491:
1460:
1396:
1385:
1384:
1374:
1356:
1347:
1336:
1335:
1325:
1307:
1296:
1285:
1258:
1235:
1233:
1210:
1169:
1141:
1136:
1134:
1125:
1120:
1117:
1093:
1088:
1086:
1077:
1072:
1050:
1015:
977:
956:
950:
926:
920:
74:
68:
59:
58:
56:
8134:For a particularly clear derivation see
8122:"Evolution of Phase Space Probabilities"
6053:
3805:
3791:compressibility of the probability fluid
871:
8142:The Principles of Statistical Mechanics
7800:
3781:of quantum mechanics, substituting the
3711:
3606:
2258:{\displaystyle ({\dot {p}},{\dot {q}})}
809:, named after the French mathematician
108:
44:
3394:a 2n-dimensional symplectic manifold,
8241:
8239:
8167:"Phase Space and Liouville's Theorem"
7716:non-equilibrium statistical mechanics
3942:particles. Within phase space, these
471:Newton's law of universal gravitation
7:
8274:"Kinetic Theory in Curved Spacetime"
4803:{\displaystyle \mathrm {d} \Gamma '}
4483:{\displaystyle {\frac {dp_{i}}{dt}}}
4439:{\displaystyle {\frac {dq_{i}}{dt}}}
3544:The analog of Liouville equation in
1010:. Then the phase space distribution
4531:{\displaystyle \mathrm {d} \Gamma }
2715:{\displaystyle (q^{\mu },p_{\mu })}
850:measure-preserving dynamical system
452:Mechanics of planar particle motion
60:
8007:The theory of open quantum systems
7764:will derive the Liouville theorem.
7107:
7103:
6922:
6917:
6761:
6748:
6734:
6711:
6698:
6684:
6537:
6513:
6491:
6467:
6321:
6313:
6258:
6250:
5613:
5598:
5594:
5580:
5563:
5558:
5544:
5512:
5475:
5471:
5325:
5320:
4993:
4969:
4947:
4923:
4793:
4788:
4722:
4698:
4618:
4594:
4525:
4521:
4129:
4100:
4096:
4082:
3986:
3982:
3950:
3921:
3585:
3577:
3548:describes the time evolution of a
3460:In fact, the symplectic structure
3281:
3277:
3229:
3225:
3177:
3173:
3155:
3147:
3125:
3121:
3103:
3095:
2996:
2992:
2974:
2966:
2944:
2940:
2922:
2914:
2586:
2572:
2564:
2496:
2492:
2474:
2466:
2444:
2440:
2422:
2414:
2364:
2303:
2295:
2154:
2127:
2097:
2070:
2000:
1987:
1973:
1950:
1936:
1922:
1865:
1841:
1819:
1795:
1633:
1600:
1578:
1545:
1504:
1496:
1367:
1359:
1318:
1310:
1269:
1261:
1137:
1121:
1089:
1073:
25:
8182:-dimensional continuity equation.
7990:"Liouville's theorem for pedants"
7163:the modified ones above. Letting
8272:Misner; Thorne; Wheeler (1973).
8097:Statistical Physics of Particles
5991:
5983:
4144:{\displaystyle \rho (\Gamma ,t)}
2594:
2372:
813:, is a key theorem in classical
756:
755:
742:
75:
7988:Nash, Oliver (8 January 2015).
7962:Geometry, Topology, and Physics
7819:Journal of Mathematical Physics
3842:and are considering the region
2652:(locally, this measure is the 6
1453:-dimensional divergence theorem
840:Liouville's theorem applies to
8337:Statistical mechanics theorems
7863:, 57–58 (1884). Reproduced in
7503:
7461:
7393:
7387:
7226:
7220:
6906:we find the following result:
6089:
6077:
6013:
6001:
5995:
5979:
5791:
5785:
5695:
5689:
5622:
5610:
5534:
5508:
4395:{\displaystyle {\dot {p_{i}}}}
4359:{\displaystyle {\dot {q_{i}}}}
4138:
4126:
3876:
3861:
3732:
3720:
3624:
3612:
3517:
3511:
3438:
3425:
3381:
3369:
3050:
3038:
2875:
2863:
2761:
2749:
2709:
2683:
2252:
2222:
1736:
1680:
1628:
1603:
1573:
1548:
1192:
1174:
1067:
1055:
1032:
1020:
1:
8332:Theorems in dynamical systems
8322:Eponymous theorems of physics
8282:. Freeman. pp. 583–590.
8009:, by Breuer and Petruccione,
6176:{\displaystyle -\gamma p_{i}}
4538:, the side lengths change as
378:Koopman–von Neumann mechanics
8120:Eastman, Peter (2014–2015).
7783:Boltzmann transport equation
7727:second law of thermodynamics
6026:with frequency dependent on
2722:of a particular Hamiltonian
1479:-dimensional version of the
1446:proof of Liouville's theorem
1198:{\displaystyle \rho (p,q;t)}
1003:{\displaystyle i=1,\dots ,n}
910:Hamiltonian dynamical system
876:Evolution of an ensemble of
446:Non-inertial reference frame
8215:"Physics 127a: Class Notes"
8071:10.1016/j.physa.2017.10.047
3835:{\displaystyle m=\omega =1}
3387:{\displaystyle (M,\omega )}
2767:{\displaystyle (M,\omega )}
2345:or, in terms of the linear
1045:determines the probability
862:internal degrees of freedom
373:Appell's equation of motion
243:Inertial frame of reference
8353:
7905:Liouville, Joseph (1838).
7752:Virtually any textbook on
6050:Damped harmonic oscillator
5082:simple harmonic oscillator
3540:Quantum Liouville equation
2774:endowed with a symplectic
1038:{\displaystyle \rho (p,q)}
29:
8145:. Dover. pp. 48–51.
7196:for convenience, we find
2208:{\displaystyle d\rho /dt}
1164:governs the evolution of
7960:Nakahara, Mikio (2003).
6899:{\displaystyle \delta t}
4833:, we get the following:
4826:{\displaystyle \delta t}
4506:{\displaystyle \delta t}
4167:{\displaystyle \delta t}
3062:{\displaystyle f(q,p)=H}
2853:generated by a function
2851:Hamiltonian vector field
536:Rotating reference frame
368:Hamilton–Jacobi equation
7892:Charles Scribner's Sons
7767:In plasma physics, the
7747:electrical conductivity
6196:{\displaystyle \gamma }
6039:{\displaystyle \omega }
5965:{\displaystyle \omega }
3882:{\displaystyle p,q\in }
3779:phase-space formulation
3473:{\displaystyle \omega }
2644:, the phase space is a
477:Newton's laws of motion
337:Newton's laws of motion
7735:transport coefficients
7699:
7679:
7652:
7622:
7190:
7149:
7058:
6953:
6900:
6874:
6576:
6364:
6205:non-conservative force
6197:
6177:
6144:
6124:
6103:
6096:
6040:
6020:
5966:
5946:
5926:
5899:
5869:
5659:
5632:
5486:
5426:
5356:
5300:
5213:
5124:
5074:
5055:
5032:
4827:
4804:
4771:
4532:
4507:
4484:
4440:
4396:
4360:
4321:
4168:
4145:
4110:
4062:
4013:
3965:
3936:
3907:
3890:
3883:
3836:
3802:SHO phase-space volume
3768:
3745:
3634:
3554:Canonical quantization
3530:
3474:
3451:
3388:
3356:
3319:
3063:
3019:
2882:
2881:{\displaystyle f(q,p)}
2829:
2768:
2736:
2716:
2618:
2542:
2405:
2336:
2259:
2209:
2171:
2114:
2057:
2034:
1912:
1786:
1743:
1664:
1536:
1469:
1417:
1301:
1219:
1199:
1154:
1106:
1039:
1004:
966:
936:
893:
504:Simple harmonic motion
417:Euler's laws of motion
211:D'Alembert's principle
94:
8327:Hamiltonian mechanics
8023:Statistical mechanics
7882:Gibbs, Josiah Willard
7758:statistical mechanics
7754:Hamiltonian mechanics
7700:
7680:
7678:{\displaystyle P_{i}}
7653:
7651:{\displaystyle Q_{i}}
7623:
7191:
7150:
7038:
6933:
6901:
6875:
6577:
6365:
6198:
6178:
6145:
6125:
6097:
6057:
6041:
6021:
5967:
5947:
5927:
5925:{\displaystyle P_{i}}
5900:
5898:{\displaystyle Q_{i}}
5870:
5660:
5633:
5487:
5406:
5336:
5301:
5214:
5101:
5075:
5056:
5033:
4828:
4805:
4772:
4533:
4508:
4485:
4441:
4397:
4361:
4322:
4169:
4146:
4111:
4063:
3993:
3966:
3937:
3908:
3884:
3837:
3809:
3769:
3746:
3662:, and takes the form
3635:
3531:
3475:
3452:
3389:
3357:
3355:{\displaystyle X_{H}}
3320:
3064:
3020:
2883:
2830:
2769:
2737:
2717:
2642:Hamiltonian mechanics
2619:
2543:
2385:
2337:
2260:
2210:
2179:convective derivative
2172:
2115:
2058:
2035:
1892:
1766:
1744:
1673:That is, the 3-tuple
1665:
1516:
1470:
1468:{\displaystyle \rho }
1418:
1281:
1220:
1200:
1155:
1107:
1040:
1005:
967:
965:{\displaystyle p_{i}}
937:
935:{\displaystyle q_{i}}
914:canonical coordinates
900:distribution function
875:
819:Hamiltonian mechanics
358:Hamiltonian mechanics
176:Statistical mechanics
95:
7743:thermal conductivity
7731:Green–Kubo relations
7689:
7662:
7635:
7203:
7167:
6913:
6887:
6592:
6384:
6214:
6187:
6154:
6134:
6130:particles each in a
6114:
6062:
6030:
5976:
5956:
5936:
5909:
5882:
5672:
5649:
5502:
5316:
5229:
5091:
5064:
5045:
4840:
4814:
4784:
4545:
4517:
4494:
4450:
4406:
4370:
4334:
4181:
4155:
4120:
4075:
3978:
3946:
3917:
3897:
3846:
3814:
3758:
3669:
3650:When applied to the
3571:
3558:von Neumann equation
3487:
3464:
3401:
3366:
3339:
3076:
3032:
2895:
2857:
2784:
2746:
2726:
2680:
2558:
2360:
2289:
2219:
2185:
2124:
2067:
2047:
1760:
1677:
1490:
1459:
1428:Hamilton's equations
1232:
1209:
1168:
1116:
1049:
1014:
976:
949:
919:
905:Josiah Willard Gibbs
890:Hamilton's equations
842:conservative systems
835:a priori probability
581:Angular acceleration
573:Rotational frequency
353:Lagrangian mechanics
346:Analytical mechanics
102:Second law of motion
55:
8063:2018PhyA..502..201O
7831:1963JMP.....4..174K
7762:symplectic geometry
7723:fluctuation theorem
6999:
6976:
6630:
6614:
6418:
6402:
5402:
5379:
5263:
5247:
5199:
5159:
4874:
4858:
4675:
4571:
4265:
4204:
3660:Ehrenfest's theorem
2847:symplectic manifold
2845:On our phase space
2674:symplectic geometry
2668:Symplectic geometry
1481:continuity equation
854:symplectic geometry
807:Liouville's theorem
433:Harmonic oscillator
411:Equations of motion
46:Classical mechanics
40:Part of a series on
32:Liouville's theorem
7695:
7675:
7648:
7618:
7616:
7186:
7145:
7144:
6987:
6964:
6896:
6870:
6868:
6618:
6602:
6572:
6406:
6390:
6360:
6358:
6193:
6173:
6140:
6120:
6104:
6092:
6036:
6016:
5962:
5942:
5922:
5895:
5865:
5863:
5655:
5628:
5482:
5481:
5390:
5367:
5296:
5251:
5235:
5209:
5208:
5185:
5145:
5070:
5051:
5028:
4862:
4846:
4823:
4800:
4767:
4762:
4663:
4559:
4528:
4503:
4480:
4436:
4392:
4356:
4317:
4312:
4253:
4192:
4164:
4141:
4106:
4058:
4057:
3961:
3932:
3903:
3891:
3879:
3832:
3764:
3741:
3630:
3526:
3470:
3447:
3384:
3352:
3315:
3059:
3015:
2878:
2825:
2764:
2732:
2712:
2614:
2538:
2347:Liouville operator
2332:
2269:Other formulations
2255:
2205:
2167:
2110:
2053:
2030:
1739:
1660:
1465:
1413:
1215:
1195:
1162:Liouville equation
1150:
1102:
1035:
1000:
962:
932:
894:
868:Liouville equation
858:chemical reactions
821:. It asserts that
749:Physics portal
363:Routhian mechanics
238:Frame of reference
90:
8106:978-0-521-87342-0
7971:978-0-7503-0606-5
7940:. pp. 48–49.
7839:10.1063/1.1703941
7698:{\displaystyle i}
7631:where the values
7585:
7563:
7368:
7184:
6775:
6725:
6551:
6532:
6505:
6486:
6335:
6297:
6272:
6237:
6143:{\displaystyle 3}
6123:{\displaystyle N}
6076:
5945:{\displaystyle i}
5752:
5658:{\displaystyle H}
5602:
5571:
5183:
5143:
5073:{\displaystyle 3}
5054:{\displaystyle N}
5007:
4988:
4961:
4942:
4736:
4717:
4632:
4613:
4478:
4434:
4389:
4353:
4298:
4237:
4104:
3906:{\displaystyle N}
3767:{\displaystyle A}
3715:
3685:
3652:expectation value
3610:
3592:
3546:quantum mechanics
3313:
3295:
3273:
3243:
3221:
3191:
3169:
3139:
3117:
3010:
2988:
2958:
2936:
2735:{\displaystyle H}
2638:dynamical systems
2601:
2579:
2510:
2488:
2458:
2436:
2379:
2310:
2249:
2234:
2140:
2083:
2056:{\displaystyle H}
2014:
1964:
1879:
1854:
1833:
1808:
1751:conserved current
1727:
1702:
1647:
1619:
1592:
1564:
1511:
1393:
1381:
1344:
1332:
1276:
1253:
1218:{\displaystyle t}
944:conjugate momenta
799:
798:
546:Centrifugal force
541:Centripetal force
497:Euler's equations
482:Relative velocity
258:Moment of inertia
88:
62:
16:(Redirected from
8344:
8308:
8293:
8268:
8250:
8243:
8234:
8228:
8226:
8224:
8219:
8211:
8205:
8203:
8201:
8199:
8189:
8183:
8177:
8175:
8173:
8163:
8157:
8156:
8132:
8126:
8125:
8117:
8111:
8110:
8089:
8083:
8082:
8056:
8036:
8030:
8020:
8014:
8004:
7998:
7996:
7994:
7985:
7976:
7975:
7957:
7951:
7948:
7942:
7941:
7939:
7928:
7922:
7921:
7911:
7902:
7896:
7895:
7878:
7872:
7857:
7851:
7850:
7814:
7808:
7805:
7704:
7702:
7701:
7696:
7684:
7682:
7681:
7676:
7674:
7673:
7657:
7655:
7654:
7649:
7647:
7646:
7627:
7625:
7624:
7619:
7617:
7610:
7606:
7605:
7604:
7586:
7581:
7580:
7571:
7564:
7562:
7561:
7549:
7544:
7543:
7533:
7531:
7527:
7526:
7522:
7521:
7502:
7501:
7483:
7482:
7473:
7472:
7454:
7450:
7449:
7433:
7432:
7418:
7417:
7386:
7385:
7369:
7367:
7366:
7354:
7353:
7344:
7339:
7338:
7328:
7326:
7322:
7321:
7317:
7316:
7300:
7299:
7287:
7283:
7282:
7266:
7265:
7251:
7250:
7219:
7218:
7195:
7193:
7192:
7187:
7185:
7177:
7154:
7152:
7151:
7146:
7140:
7136:
7106:
7098:
7097:
7088:
7087:
7078:
7077:
7068:
7067:
7057:
7052:
7037:
7036:
7028:
7024:
6995:
6986:
6985:
6972:
6963:
6962:
6952:
6947:
6928:
6920:
6905:
6903:
6902:
6897:
6879:
6877:
6876:
6871:
6869:
6862:
6858:
6837:
6836:
6824:
6823:
6805:
6798:
6794:
6787:
6783:
6776:
6774:
6773:
6772:
6760:
6759:
6746:
6742:
6741:
6731:
6726:
6724:
6723:
6722:
6710:
6709:
6696:
6692:
6691:
6681:
6663:
6662:
6650:
6649:
6626:
6610:
6581:
6579:
6578:
6573:
6568:
6564:
6557:
6553:
6552:
6550:
6549:
6548:
6535:
6534:
6533:
6528:
6527:
6518:
6511:
6506:
6504:
6503:
6502:
6489:
6488:
6487:
6482:
6481:
6472:
6465:
6447:
6446:
6434:
6433:
6414:
6398:
6369:
6367:
6366:
6361:
6359:
6352:
6351:
6336:
6334:
6333:
6332:
6319:
6311:
6299:
6298:
6293:
6292:
6283:
6273:
6271:
6270:
6269:
6256:
6248:
6239:
6238:
6233:
6232:
6223:
6202:
6200:
6199:
6194:
6182:
6180:
6179:
6174:
6172:
6171:
6149:
6147:
6146:
6141:
6129:
6127:
6126:
6121:
6101:
6099:
6098:
6093:
6074:
6045:
6043:
6042:
6037:
6025:
6023:
6022:
6017:
5994:
5986:
5971:
5969:
5968:
5963:
5951:
5949:
5948:
5943:
5931:
5929:
5928:
5923:
5921:
5920:
5904:
5902:
5901:
5896:
5894:
5893:
5874:
5872:
5871:
5866:
5864:
5857:
5843:
5842:
5824:
5810:
5809:
5784:
5783:
5767:
5753:
5751:
5743:
5742:
5733:
5728:
5714:
5713:
5688:
5687:
5664:
5662:
5661:
5656:
5637:
5635:
5634:
5629:
5603:
5601:
5597:
5591:
5590:
5589:
5583:
5577:
5572:
5570:
5569:
5561:
5555:
5554:
5553:
5547:
5541:
5518:
5491:
5489:
5488:
5483:
5474:
5466:
5465:
5456:
5455:
5446:
5445:
5436:
5435:
5425:
5420:
5398:
5389:
5388:
5375:
5366:
5365:
5355:
5350:
5331:
5323:
5305:
5303:
5302:
5297:
5292:
5291:
5279:
5278:
5259:
5243:
5218:
5216:
5215:
5210:
5204:
5200:
5198:
5193:
5184:
5179:
5178:
5177:
5164:
5158:
5153:
5144:
5142:
5131:
5123:
5115:
5079:
5077:
5076:
5071:
5060:
5058:
5057:
5052:
5037:
5035:
5034:
5029:
5024:
5020:
5013:
5009:
5008:
5006:
5005:
5004:
4991:
4990:
4989:
4984:
4983:
4974:
4967:
4962:
4960:
4959:
4958:
4945:
4944:
4943:
4938:
4937:
4928:
4921:
4903:
4902:
4890:
4889:
4870:
4854:
4832:
4830:
4829:
4824:
4809:
4807:
4806:
4801:
4799:
4791:
4776:
4774:
4773:
4768:
4766:
4765:
4750:
4749:
4737:
4735:
4734:
4733:
4720:
4719:
4718:
4713:
4712:
4703:
4696:
4691:
4690:
4671:
4646:
4645:
4633:
4631:
4630:
4629:
4616:
4615:
4614:
4609:
4608:
4599:
4592:
4587:
4586:
4567:
4537:
4535:
4534:
4529:
4524:
4512:
4510:
4509:
4504:
4489:
4487:
4486:
4481:
4479:
4477:
4469:
4468:
4467:
4454:
4445:
4443:
4442:
4437:
4435:
4433:
4425:
4424:
4423:
4410:
4401:
4399:
4398:
4393:
4391:
4390:
4385:
4384:
4375:
4365:
4363:
4362:
4357:
4355:
4354:
4349:
4348:
4339:
4326:
4324:
4323:
4318:
4316:
4315:
4300:
4299:
4294:
4293:
4284:
4278:
4277:
4261:
4239:
4238:
4233:
4232:
4223:
4217:
4216:
4200:
4173:
4171:
4170:
4165:
4150:
4148:
4147:
4142:
4115:
4113:
4112:
4107:
4105:
4103:
4099:
4093:
4092:
4091:
4085:
4079:
4067:
4065:
4064:
4059:
4053:
4052:
4043:
4042:
4033:
4032:
4023:
4022:
4012:
4007:
3985:
3970:
3968:
3967:
3962:
3960:
3959:
3953:
3941:
3939:
3938:
3933:
3931:
3930:
3924:
3912:
3910:
3909:
3904:
3888:
3886:
3885:
3880:
3841:
3839:
3838:
3833:
3787:Poisson brackets
3773:
3771:
3770:
3765:
3750:
3748:
3747:
3742:
3716:
3714:
3703:
3686:
3684:
3673:
3639:
3637:
3636:
3631:
3611:
3609:
3598:
3593:
3591:
3583:
3575:
3535:
3533:
3532:
3527:
3510:
3509:
3508:
3507:
3497:
3496:
3479:
3477:
3476:
3471:
3456:
3454:
3453:
3448:
3437:
3436:
3424:
3423:
3422:
3421:
3411:
3410:
3393:
3391:
3390:
3385:
3361:
3359:
3358:
3353:
3351:
3350:
3324:
3322:
3321:
3316:
3314:
3312:
3301:
3296:
3294:
3293:
3292:
3276:
3274:
3272:
3264:
3263:
3262:
3249:
3244:
3242:
3241:
3240:
3224:
3222:
3220:
3212:
3211:
3210:
3197:
3192:
3190:
3189:
3188:
3172:
3170:
3168:
3167:
3166:
3153:
3145:
3140:
3138:
3137:
3136:
3120:
3118:
3116:
3115:
3114:
3101:
3093:
3088:
3087:
3068:
3066:
3065:
3060:
3024:
3022:
3021:
3016:
3011:
3009:
3008:
3007:
2991:
2989:
2987:
2986:
2985:
2972:
2964:
2959:
2957:
2956:
2955:
2939:
2937:
2935:
2934:
2933:
2920:
2912:
2907:
2906:
2887:
2885:
2884:
2879:
2849:we can define a
2834:
2832:
2831:
2826:
2821:
2820:
2805:
2804:
2773:
2771:
2770:
2765:
2741:
2739:
2738:
2733:
2721:
2719:
2718:
2713:
2708:
2707:
2695:
2694:
2662:Hamiltonian flow
2658:Lebesgue measure
2623:
2621:
2620:
2615:
2604:
2603:
2602:
2597:
2592:
2589:
2580:
2578:
2570:
2562:
2547:
2545:
2544:
2539:
2516:
2512:
2511:
2509:
2508:
2507:
2491:
2489:
2487:
2486:
2485:
2472:
2464:
2459:
2457:
2456:
2455:
2439:
2437:
2435:
2434:
2433:
2420:
2412:
2404:
2399:
2381:
2380:
2375:
2370:
2367:
2341:
2339:
2338:
2333:
2311:
2309:
2301:
2293:
2264:
2262:
2261:
2256:
2251:
2250:
2242:
2236:
2235:
2227:
2214:
2212:
2211:
2206:
2198:
2181:of the density,
2176:
2174:
2173:
2168:
2166:
2165:
2153:
2148:
2147:
2142:
2141:
2133:
2119:
2117:
2116:
2111:
2109:
2108:
2096:
2091:
2090:
2085:
2084:
2076:
2062:
2060:
2059:
2054:
2039:
2037:
2036:
2031:
2020:
2016:
2015:
2013:
2012:
2011:
1999:
1998:
1985:
1981:
1980:
1970:
1965:
1963:
1962:
1961:
1948:
1947:
1934:
1930:
1929:
1919:
1911:
1906:
1885:
1881:
1880:
1878:
1877:
1876:
1863:
1862:
1861:
1856:
1855:
1847:
1839:
1834:
1832:
1831:
1830:
1817:
1816:
1815:
1810:
1809:
1801:
1793:
1785:
1780:
1748:
1746:
1745:
1740:
1735:
1734:
1729:
1728:
1720:
1710:
1709:
1704:
1703:
1695:
1669:
1667:
1666:
1661:
1653:
1649:
1648:
1646:
1645:
1644:
1631:
1627:
1626:
1621:
1620:
1612:
1598:
1593:
1591:
1590:
1589:
1576:
1572:
1571:
1566:
1565:
1557:
1543:
1535:
1530:
1512:
1510:
1502:
1494:
1474:
1472:
1471:
1466:
1422:
1420:
1419:
1414:
1406:
1402:
1401:
1400:
1395:
1394:
1386:
1382:
1380:
1379:
1378:
1365:
1357:
1352:
1351:
1346:
1345:
1337:
1333:
1331:
1330:
1329:
1316:
1308:
1300:
1295:
1277:
1275:
1267:
1259:
1254:
1252:
1244:
1236:
1224:
1222:
1221:
1216:
1204:
1202:
1201:
1196:
1159:
1157:
1156:
1151:
1146:
1145:
1140:
1130:
1129:
1124:
1111:
1109:
1108:
1103:
1098:
1097:
1092:
1082:
1081:
1076:
1044:
1042:
1041:
1036:
1009:
1007:
1006:
1001:
971:
969:
968:
963:
961:
960:
941:
939:
938:
933:
931:
930:
811:Joseph Liouville
791:
784:
777:
764:
759:
758:
751:
747:
746:
652:Johann Bernoulli
647:Daniel Bernoulli
568:Tangential speed
472:
448:
423:Fictitious force
418:
270:Mechanical power
260:
201:Angular momentum
99:
97:
96:
91:
89:
87:
79:
78:
69:
64:
63:
37:
21:
8352:
8351:
8347:
8346:
8345:
8343:
8342:
8341:
8312:
8311:
8303:
8300:
8290:
8271:
8263:Murugeshan, R.
8262:
8259:
8257:Further reading
8254:
8253:
8244:
8237:
8222:
8220:
8217:
8213:
8212:
8208:
8197:
8195:
8191:
8190:
8186:
8171:
8169:
8165:
8164:
8160:
8153:
8135:
8133:
8129:
8119:
8118:
8114:
8107:
8091:
8090:
8086:
8038:
8037:
8033:
8021:
8017:
8005:
8001:
7992:
7987:
7986:
7979:
7972:
7959:
7958:
7954:
7949:
7945:
7937:
7930:
7929:
7925:
7909:
7904:
7903:
7899:
7880:
7879:
7875:
7858:
7854:
7816:
7815:
7811:
7806:
7802:
7797:
7779:
7769:Vlasov equation
7725:from which the
7711:
7687:
7686:
7665:
7660:
7659:
7638:
7633:
7632:
7615:
7614:
7596:
7572:
7569:
7565:
7553:
7535:
7532:
7513:
7493:
7474:
7464:
7441:
7424:
7423:
7419:
7403:
7396:
7377:
7374:
7373:
7358:
7345:
7330:
7327:
7308:
7291:
7274:
7257:
7256:
7252:
7236:
7229:
7210:
7201:
7200:
7165:
7164:
7114:
7110:
7089:
7079:
7069:
7059:
7008:
7004:
7003:
6977:
6954:
6921:
6911:
6910:
6885:
6884:
6867:
6866:
6842:
6838:
6828:
6815:
6803:
6802:
6764:
6751:
6747:
6733:
6732:
6714:
6701:
6697:
6683:
6682:
6679:
6675:
6668:
6664:
6654:
6641:
6631:
6590:
6589:
6540:
6536:
6519:
6512:
6494:
6490:
6473:
6466:
6463:
6459:
6452:
6448:
6438:
6425:
6382:
6381:
6357:
6356:
6343:
6324:
6320:
6312:
6300:
6284:
6278:
6277:
6261:
6257:
6249:
6240:
6224:
6212:
6211:
6185:
6184:
6163:
6152:
6151:
6132:
6131:
6112:
6111:
6060:
6059:
6052:
6028:
6027:
5974:
5973:
5954:
5953:
5934:
5933:
5912:
5907:
5906:
5885:
5880:
5879:
5862:
5861:
5834:
5801:
5794:
5775:
5772:
5771:
5744:
5734:
5705:
5698:
5679:
5670:
5669:
5647:
5646:
5592:
5578:
5562:
5556:
5542:
5511:
5500:
5499:
5457:
5447:
5437:
5427:
5380:
5357:
5324:
5314:
5313:
5283:
5270:
5227:
5226:
5169:
5165:
5135:
5129:
5125:
5089:
5088:
5062:
5061:
5043:
5042:
4996:
4992:
4975:
4968:
4950:
4946:
4929:
4922:
4919:
4915:
4908:
4904:
4894:
4881:
4838:
4837:
4812:
4811:
4792:
4782:
4781:
4761:
4760:
4741:
4725:
4721:
4704:
4697:
4682:
4657:
4656:
4637:
4621:
4617:
4600:
4593:
4578:
4549:
4543:
4542:
4515:
4514:
4492:
4491:
4470:
4459:
4455:
4448:
4447:
4426:
4415:
4411:
4404:
4403:
4376:
4368:
4367:
4340:
4332:
4331:
4311:
4310:
4285:
4269:
4250:
4249:
4224:
4208:
4185:
4179:
4178:
4153:
4152:
4118:
4117:
4094:
4080:
4073:
4072:
4044:
4034:
4024:
4014:
3976:
3975:
3944:
3943:
3915:
3914:
3895:
3894:
3844:
3843:
3812:
3811:
3804:
3799:
3756:
3755:
3707:
3677:
3667:
3666:
3643:where ρ is the
3602:
3584:
3576:
3569:
3568:
3542:
3499:
3490:
3485:
3484:
3462:
3461:
3428:
3413:
3404:
3399:
3398:
3364:
3363:
3362:. That is, for
3342:
3337:
3336:
3305:
3284:
3280:
3265:
3254:
3250:
3232:
3228:
3213:
3202:
3198:
3180:
3176:
3158:
3154:
3146:
3128:
3124:
3106:
3102:
3094:
3079:
3074:
3073:
3030:
3029:
2999:
2995:
2977:
2973:
2965:
2947:
2943:
2925:
2921:
2913:
2898:
2893:
2892:
2855:
2854:
2812:
2796:
2782:
2781:
2744:
2743:
2724:
2723:
2699:
2686:
2678:
2677:
2670:
2646:smooth manifold
2630:
2571:
2563:
2556:
2555:
2499:
2495:
2477:
2473:
2465:
2447:
2443:
2425:
2421:
2413:
2410:
2406:
2358:
2357:
2302:
2294:
2287:
2286:
2280:Poisson bracket
2276:
2274:Poisson bracket
2271:
2217:
2216:
2183:
2182:
2157:
2130:
2122:
2121:
2100:
2073:
2065:
2064:
2045:
2044:
2003:
1990:
1986:
1972:
1971:
1953:
1939:
1935:
1921:
1920:
1917:
1913:
1868:
1864:
1844:
1840:
1822:
1818:
1798:
1794:
1791:
1787:
1758:
1757:
1717:
1692:
1675:
1674:
1636:
1632:
1609:
1599:
1581:
1577:
1554:
1544:
1541:
1537:
1503:
1495:
1488:
1487:
1457:
1456:
1383:
1370:
1366:
1358:
1334:
1321:
1317:
1309:
1306:
1302:
1268:
1260:
1245:
1237:
1230:
1229:
1207:
1206:
1166:
1165:
1135:
1119:
1114:
1113:
1087:
1071:
1047:
1046:
1012:
1011:
974:
973:
952:
947:
946:
922:
917:
916:
870:
795:
754:
741:
740:
733:
732:
731:
606:
598:
597:
577:
531:Circular motion
525:
515:
514:
513:
470:
440:
437:
416:
395:
387:
386:
383:
382:
340:
330:
322:
321:
320:
279:
275:Mechanical work
268:
252:
190:
182:
181:
180:
135:
127:
104:
80:
70:
53:
52:
35:
28:
23:
22:
15:
12:
11:
5:
8350:
8348:
8340:
8339:
8334:
8329:
8324:
8314:
8313:
8310:
8309:
8299:
8298:External links
8296:
8295:
8294:
8288:
8269:
8265:Modern Physics
8258:
8255:
8252:
8251:
8235:
8206:
8184:
8158:
8151:
8127:
8112:
8105:
8093:Kardar, Mehran
8084:
8031:
8025:, by Schwabl,
8015:
7999:
7977:
7970:
7952:
7943:
7923:
7897:
7873:
7852:
7825:(2): 174–183.
7809:
7799:
7798:
7796:
7793:
7792:
7791:
7785:
7778:
7775:
7774:
7773:
7765:
7750:
7737:such as shear
7719:
7710:
7707:
7694:
7672:
7668:
7645:
7641:
7629:
7628:
7613:
7609:
7603:
7599:
7595:
7592:
7589:
7584:
7579:
7575:
7568:
7560:
7556:
7552:
7547:
7542:
7538:
7534:
7530:
7525:
7520:
7516:
7511:
7508:
7505:
7500:
7496:
7492:
7489:
7486:
7481:
7477:
7471:
7467:
7463:
7460:
7457:
7453:
7448:
7444:
7439:
7436:
7431:
7427:
7422:
7416:
7413:
7410:
7406:
7402:
7399:
7397:
7395:
7392:
7389:
7384:
7380:
7376:
7375:
7372:
7365:
7361:
7357:
7352:
7348:
7342:
7337:
7333:
7329:
7325:
7320:
7315:
7311:
7306:
7303:
7298:
7294:
7290:
7286:
7281:
7277:
7272:
7269:
7264:
7260:
7255:
7249:
7246:
7243:
7239:
7235:
7232:
7230:
7228:
7225:
7222:
7217:
7213:
7209:
7208:
7183:
7180:
7175:
7172:
7156:
7155:
7143:
7139:
7135:
7132:
7129:
7126:
7123:
7120:
7117:
7113:
7109:
7105:
7101:
7096:
7092:
7086:
7082:
7076:
7072:
7066:
7062:
7056:
7051:
7048:
7045:
7041:
7035:
7032:
7027:
7023:
7020:
7017:
7014:
7011:
7007:
7002:
6998:
6994:
6990:
6984:
6980:
6975:
6971:
6967:
6961:
6957:
6951:
6946:
6943:
6940:
6936:
6931:
6927:
6924:
6919:
6895:
6892:
6881:
6880:
6865:
6861:
6857:
6854:
6851:
6848:
6845:
6841:
6835:
6831:
6827:
6822:
6818:
6814:
6811:
6808:
6806:
6804:
6801:
6797:
6793:
6790:
6786:
6782:
6779:
6771:
6767:
6763:
6758:
6754:
6750:
6745:
6740:
6736:
6729:
6721:
6717:
6713:
6708:
6704:
6700:
6695:
6690:
6686:
6678:
6674:
6671:
6667:
6661:
6657:
6653:
6648:
6644:
6640:
6637:
6634:
6632:
6629:
6625:
6621:
6617:
6613:
6609:
6605:
6601:
6598:
6597:
6583:
6582:
6571:
6567:
6563:
6560:
6556:
6547:
6543:
6539:
6531:
6526:
6522:
6515:
6509:
6501:
6497:
6493:
6485:
6480:
6476:
6469:
6462:
6458:
6455:
6451:
6445:
6441:
6437:
6432:
6428:
6424:
6421:
6417:
6413:
6409:
6405:
6401:
6397:
6393:
6389:
6371:
6370:
6355:
6350:
6346:
6342:
6339:
6331:
6327:
6323:
6318:
6315:
6309:
6306:
6303:
6301:
6296:
6291:
6287:
6280:
6279:
6276:
6268:
6264:
6260:
6255:
6252:
6246:
6243:
6241:
6236:
6231:
6227:
6220:
6219:
6192:
6170:
6166:
6162:
6159:
6139:
6119:
6091:
6088:
6085:
6082:
6079:
6073:
6070:
6067:
6051:
6048:
6035:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5993:
5989:
5985:
5981:
5961:
5941:
5919:
5915:
5892:
5888:
5876:
5875:
5860:
5856:
5853:
5849:
5846:
5841:
5837:
5833:
5830:
5827:
5823:
5820:
5816:
5813:
5808:
5804:
5800:
5797:
5795:
5793:
5790:
5787:
5782:
5778:
5774:
5773:
5770:
5766:
5763:
5759:
5756:
5750:
5747:
5741:
5737:
5731:
5727:
5724:
5720:
5717:
5712:
5708:
5704:
5701:
5699:
5697:
5694:
5691:
5686:
5682:
5678:
5677:
5654:
5639:
5638:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5600:
5596:
5588:
5582:
5575:
5568:
5565:
5560:
5552:
5546:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5517:
5514:
5510:
5507:
5493:
5492:
5480:
5477:
5473:
5469:
5464:
5460:
5454:
5450:
5444:
5440:
5434:
5430:
5424:
5419:
5416:
5413:
5409:
5405:
5401:
5397:
5393:
5387:
5383:
5378:
5374:
5370:
5364:
5360:
5354:
5349:
5346:
5343:
5339:
5334:
5330:
5327:
5322:
5307:
5306:
5295:
5290:
5286:
5282:
5277:
5273:
5269:
5266:
5262:
5258:
5254:
5250:
5246:
5242:
5238:
5234:
5220:
5219:
5207:
5203:
5197:
5192:
5188:
5182:
5176:
5172:
5168:
5162:
5157:
5152:
5148:
5141:
5138:
5134:
5128:
5122:
5119:
5114:
5111:
5108:
5104:
5099:
5096:
5069:
5050:
5039:
5038:
5027:
5023:
5019:
5016:
5012:
5003:
4999:
4995:
4987:
4982:
4978:
4971:
4965:
4957:
4953:
4949:
4941:
4936:
4932:
4925:
4918:
4914:
4911:
4907:
4901:
4897:
4893:
4888:
4884:
4880:
4877:
4873:
4869:
4865:
4861:
4857:
4853:
4849:
4845:
4822:
4819:
4798:
4795:
4790:
4778:
4777:
4764:
4759:
4756:
4753:
4748:
4744:
4740:
4732:
4728:
4724:
4716:
4711:
4707:
4700:
4694:
4689:
4685:
4681:
4678:
4674:
4670:
4666:
4662:
4659:
4658:
4655:
4652:
4649:
4644:
4640:
4636:
4628:
4624:
4620:
4612:
4607:
4603:
4596:
4590:
4585:
4581:
4577:
4574:
4570:
4566:
4562:
4558:
4555:
4554:
4552:
4527:
4523:
4502:
4499:
4476:
4473:
4466:
4462:
4458:
4432:
4429:
4422:
4418:
4414:
4388:
4383:
4379:
4352:
4347:
4343:
4328:
4327:
4314:
4309:
4306:
4303:
4297:
4292:
4288:
4281:
4276:
4272:
4268:
4264:
4260:
4256:
4252:
4251:
4248:
4245:
4242:
4236:
4231:
4227:
4220:
4215:
4211:
4207:
4203:
4199:
4195:
4191:
4190:
4188:
4163:
4160:
4140:
4137:
4134:
4131:
4128:
4125:
4102:
4098:
4090:
4084:
4069:
4068:
4056:
4051:
4047:
4041:
4037:
4031:
4027:
4021:
4017:
4011:
4006:
4003:
4000:
3996:
3991:
3988:
3984:
3958:
3952:
3929:
3923:
3902:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3831:
3828:
3825:
3822:
3819:
3803:
3800:
3798:
3795:
3783:Moyal brackets
3763:
3752:
3751:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3713:
3710:
3706:
3701:
3698:
3695:
3692:
3689:
3683:
3680:
3676:
3645:density matrix
3641:
3640:
3629:
3626:
3623:
3620:
3617:
3614:
3608:
3605:
3601:
3596:
3590:
3587:
3582:
3579:
3541:
3538:
3537:
3536:
3525:
3522:
3519:
3516:
3513:
3506:
3502:
3495:
3469:
3458:
3457:
3446:
3443:
3440:
3435:
3431:
3427:
3420:
3416:
3409:
3383:
3380:
3377:
3374:
3371:
3349:
3345:
3333:Lie derivative
3326:
3325:
3311:
3308:
3304:
3299:
3291:
3287:
3283:
3279:
3271:
3268:
3261:
3257:
3253:
3247:
3239:
3235:
3231:
3227:
3219:
3216:
3209:
3205:
3201:
3195:
3187:
3183:
3179:
3175:
3165:
3161:
3157:
3152:
3149:
3143:
3135:
3131:
3127:
3123:
3113:
3109:
3105:
3100:
3097:
3091:
3086:
3082:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3026:
3025:
3014:
3006:
3002:
2998:
2994:
2984:
2980:
2976:
2971:
2968:
2962:
2954:
2950:
2946:
2942:
2932:
2928:
2924:
2919:
2916:
2910:
2905:
2901:
2877:
2874:
2871:
2868:
2865:
2862:
2840:exterior power
2836:
2835:
2824:
2819:
2815:
2811:
2808:
2803:
2799:
2795:
2792:
2789:
2763:
2760:
2757:
2754:
2751:
2742:as a manifold
2731:
2711:
2706:
2702:
2698:
2693:
2689:
2685:
2669:
2666:
2634:ergodic theory
2629:
2628:Ergodic theory
2626:
2625:
2624:
2613:
2610:
2607:
2600:
2596:
2588:
2583:
2577:
2574:
2569:
2566:
2549:
2548:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2515:
2506:
2502:
2498:
2494:
2484:
2480:
2476:
2471:
2468:
2462:
2454:
2450:
2446:
2442:
2432:
2428:
2424:
2419:
2416:
2409:
2403:
2398:
2395:
2392:
2388:
2384:
2378:
2374:
2366:
2343:
2342:
2331:
2327:
2324:
2321:
2317:
2314:
2308:
2305:
2300:
2297:
2275:
2272:
2270:
2267:
2254:
2248:
2245:
2239:
2233:
2230:
2224:
2204:
2201:
2197:
2193:
2190:
2164:
2160:
2156:
2152:
2146:
2139:
2136:
2129:
2107:
2103:
2099:
2095:
2089:
2082:
2079:
2072:
2052:
2041:
2040:
2029:
2026:
2023:
2019:
2010:
2006:
2002:
1997:
1993:
1989:
1984:
1979:
1975:
1968:
1960:
1956:
1952:
1946:
1942:
1938:
1933:
1928:
1924:
1916:
1910:
1905:
1902:
1899:
1895:
1891:
1888:
1884:
1875:
1871:
1867:
1860:
1853:
1850:
1843:
1837:
1829:
1825:
1821:
1814:
1807:
1804:
1797:
1790:
1784:
1779:
1776:
1773:
1769:
1765:
1738:
1733:
1726:
1723:
1716:
1713:
1708:
1701:
1698:
1691:
1688:
1685:
1682:
1671:
1670:
1659:
1656:
1652:
1643:
1639:
1635:
1630:
1625:
1618:
1615:
1608:
1605:
1602:
1596:
1588:
1584:
1580:
1575:
1570:
1563:
1560:
1553:
1550:
1547:
1540:
1534:
1529:
1526:
1523:
1519:
1515:
1509:
1506:
1501:
1498:
1464:
1442:
1441:
1424:
1423:
1412:
1409:
1405:
1399:
1392:
1389:
1377:
1373:
1369:
1364:
1361:
1355:
1350:
1343:
1340:
1328:
1324:
1320:
1315:
1312:
1305:
1299:
1294:
1291:
1288:
1284:
1280:
1274:
1271:
1266:
1263:
1257:
1251:
1248:
1243:
1240:
1214:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1149:
1144:
1139:
1133:
1128:
1123:
1101:
1096:
1091:
1085:
1080:
1075:
1069:
1066:
1063:
1060:
1057:
1054:
1034:
1031:
1028:
1025:
1022:
1019:
999:
996:
993:
990:
987:
984:
981:
959:
955:
929:
925:
886:potential well
869:
866:
797:
796:
794:
793:
786:
779:
771:
768:
767:
766:
765:
752:
735:
734:
730:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
608:
607:
604:
603:
600:
599:
596:
595:
576:
575:
570:
565:
560:
558:Coriolis force
555:
554:
553:
543:
538:
533:
527:
526:
521:
520:
517:
516:
512:
511:
506:
501:
500:
499:
494:
484:
479:
474:
467:
456:
455:
454:
449:
436:
435:
430:
425:
420:
413:
408:
403:
397:
396:
393:
392:
389:
388:
385:
384:
381:
380:
375:
370:
365:
360:
355:
349:
343:
341:
334:
331:
328:
327:
324:
323:
319:
318:
313:
308:
303:
298:
293:
288:
283:
277:
272:
266:
261:
250:
245:
240:
235:
230:
229:
228:
223:
213:
208:
203:
198:
192:
191:
188:
187:
184:
183:
179:
178:
173:
168:
163:
158:
153:
148:
143:
137:
136:
133:
132:
129:
128:
126:
125:
120:
115:
109:
106:
105:
100:
86:
83:
77:
73:
67:
49:
48:
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8349:
8338:
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8319:
8317:
8306:
8302:
8301:
8297:
8291:
8289:9781400889099
8285:
8281:
8280:
8275:
8270:
8266:
8261:
8260:
8256:
8248:
8242:
8240:
8236:
8232:
8216:
8210:
8207:
8194:
8188:
8185:
8181:
8168:
8162:
8159:
8154:
8152:9780486638966
8148:
8144:
8143:
8138:
8137:Tolman, R. C.
8131:
8128:
8123:
8116:
8113:
8108:
8102:
8098:
8094:
8088:
8085:
8080:
8076:
8072:
8068:
8064:
8060:
8055:
8050:
8046:
8042:
8035:
8032:
8028:
8024:
8019:
8016:
8012:
8008:
8003:
8000:
7991:
7984:
7982:
7978:
7973:
7967:
7963:
7956:
7953:
7947:
7944:
7936:
7935:
7927:
7924:
7919:
7915:
7908:
7901:
7898:
7893:
7889:
7888:
7883:
7877:
7874:
7870:
7866:
7862:
7856:
7853:
7848:
7844:
7840:
7836:
7832:
7828:
7824:
7820:
7813:
7810:
7804:
7801:
7794:
7789:
7786:
7784:
7781:
7780:
7776:
7770:
7766:
7763:
7759:
7755:
7751:
7748:
7744:
7740:
7736:
7732:
7728:
7724:
7720:
7717:
7713:
7712:
7708:
7706:
7692:
7670:
7666:
7643:
7639:
7611:
7607:
7601:
7597:
7593:
7590:
7587:
7582:
7577:
7573:
7566:
7558:
7554:
7550:
7545:
7540:
7536:
7528:
7523:
7518:
7514:
7509:
7506:
7498:
7494:
7490:
7487:
7484:
7479:
7475:
7469:
7465:
7458:
7455:
7451:
7446:
7442:
7437:
7434:
7429:
7425:
7420:
7414:
7411:
7408:
7404:
7400:
7398:
7390:
7382:
7378:
7370:
7363:
7359:
7355:
7350:
7346:
7340:
7335:
7331:
7323:
7318:
7313:
7309:
7304:
7301:
7296:
7292:
7288:
7284:
7279:
7275:
7270:
7267:
7262:
7258:
7253:
7247:
7244:
7241:
7237:
7233:
7231:
7223:
7215:
7211:
7199:
7198:
7197:
7181:
7178:
7173:
7170:
7160:
7141:
7137:
7133:
7130:
7127:
7124:
7121:
7118:
7115:
7111:
7099:
7094:
7090:
7084:
7080:
7074:
7070:
7064:
7060:
7054:
7049:
7046:
7043:
7039:
7033:
7030:
7025:
7021:
7018:
7015:
7012:
7009:
7005:
7000:
6996:
6992:
6988:
6982:
6978:
6973:
6969:
6965:
6959:
6955:
6949:
6944:
6941:
6938:
6934:
6929:
6925:
6909:
6908:
6907:
6893:
6890:
6863:
6859:
6855:
6852:
6849:
6846:
6843:
6839:
6833:
6829:
6825:
6820:
6816:
6812:
6809:
6807:
6799:
6795:
6791:
6788:
6784:
6780:
6777:
6769:
6765:
6756:
6752:
6743:
6738:
6727:
6719:
6715:
6706:
6702:
6693:
6688:
6676:
6672:
6669:
6665:
6659:
6655:
6651:
6646:
6642:
6638:
6635:
6633:
6627:
6623:
6619:
6615:
6611:
6607:
6603:
6599:
6588:
6587:
6586:
6569:
6565:
6561:
6558:
6554:
6545:
6541:
6529:
6524:
6520:
6507:
6499:
6495:
6483:
6478:
6474:
6460:
6456:
6453:
6449:
6443:
6439:
6435:
6430:
6426:
6422:
6419:
6415:
6411:
6407:
6403:
6399:
6395:
6391:
6387:
6380:
6379:
6378:
6375:
6353:
6348:
6344:
6340:
6337:
6329:
6325:
6316:
6307:
6304:
6302:
6294:
6289:
6285:
6274:
6266:
6262:
6253:
6244:
6242:
6234:
6229:
6225:
6210:
6209:
6208:
6206:
6190:
6168:
6164:
6160:
6157:
6137:
6117:
6109:
6086:
6083:
6080:
6071:
6068:
6065:
6056:
6049:
6047:
6033:
6010:
6007:
6004:
5998:
5987:
5959:
5939:
5917:
5913:
5890:
5886:
5858:
5854:
5851:
5847:
5844:
5839:
5835:
5831:
5828:
5825:
5821:
5818:
5814:
5811:
5806:
5802:
5798:
5796:
5788:
5780:
5776:
5768:
5764:
5761:
5757:
5754:
5748:
5745:
5739:
5735:
5729:
5725:
5722:
5718:
5715:
5710:
5706:
5702:
5700:
5692:
5684:
5680:
5668:
5667:
5666:
5652:
5642:
5625:
5619:
5616:
5607:
5604:
5573:
5566:
5537:
5531:
5528:
5525:
5522:
5519:
5515:
5505:
5498:
5497:
5496:
5478:
5467:
5462:
5458:
5452:
5448:
5442:
5438:
5432:
5428:
5422:
5417:
5414:
5411:
5407:
5403:
5399:
5395:
5391:
5385:
5381:
5376:
5372:
5368:
5362:
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5352:
5347:
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5332:
5328:
5312:
5311:
5310:
5293:
5288:
5284:
5280:
5275:
5271:
5267:
5264:
5260:
5256:
5252:
5248:
5244:
5240:
5236:
5232:
5225:
5224:
5223:
5205:
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5190:
5186:
5180:
5174:
5170:
5166:
5160:
5155:
5150:
5146:
5139:
5136:
5132:
5126:
5120:
5117:
5112:
5109:
5106:
5102:
5097:
5094:
5087:
5086:
5085:
5083:
5067:
5048:
5025:
5021:
5017:
5014:
5010:
5001:
4997:
4985:
4980:
4976:
4963:
4955:
4951:
4939:
4934:
4930:
4916:
4912:
4909:
4905:
4899:
4895:
4891:
4886:
4882:
4878:
4875:
4871:
4867:
4863:
4859:
4855:
4851:
4847:
4843:
4836:
4835:
4834:
4820:
4817:
4796:
4757:
4754:
4751:
4746:
4742:
4738:
4730:
4726:
4714:
4709:
4705:
4692:
4687:
4683:
4679:
4676:
4672:
4668:
4664:
4660:
4653:
4650:
4647:
4642:
4638:
4634:
4626:
4622:
4610:
4605:
4601:
4588:
4583:
4579:
4575:
4572:
4568:
4564:
4560:
4556:
4550:
4541:
4540:
4539:
4500:
4497:
4474:
4471:
4464:
4460:
4456:
4430:
4427:
4420:
4416:
4412:
4386:
4381:
4377:
4350:
4345:
4341:
4307:
4304:
4301:
4295:
4290:
4286:
4279:
4274:
4270:
4266:
4262:
4258:
4254:
4246:
4243:
4240:
4234:
4229:
4225:
4218:
4213:
4209:
4205:
4201:
4197:
4193:
4186:
4177:
4176:
4175:
4161:
4158:
4135:
4132:
4123:
4054:
4049:
4045:
4039:
4035:
4029:
4025:
4019:
4015:
4009:
4004:
4001:
3998:
3994:
3989:
3974:
3973:
3972:
3900:
3873:
3870:
3867:
3864:
3858:
3855:
3852:
3849:
3829:
3826:
3823:
3820:
3817:
3808:
3801:
3796:
3794:
3792:
3788:
3784:
3780:
3775:
3761:
3738:
3729:
3726:
3723:
3708:
3704:
3699:
3696:
3690:
3681:
3678:
3674:
3665:
3664:
3663:
3661:
3657:
3653:
3648:
3646:
3627:
3621:
3618:
3615:
3603:
3599:
3594:
3588:
3580:
3567:
3566:
3565:
3563:
3559:
3555:
3551:
3547:
3539:
3523:
3520:
3514:
3504:
3500:
3483:
3482:
3481:
3467:
3444:
3441:
3433:
3429:
3418:
3414:
3397:
3396:
3395:
3378:
3375:
3372:
3347:
3343:
3334:
3329:
3309:
3306:
3302:
3297:
3289:
3285:
3269:
3266:
3259:
3255:
3251:
3245:
3237:
3233:
3217:
3214:
3207:
3203:
3199:
3193:
3185:
3181:
3163:
3159:
3150:
3141:
3133:
3129:
3111:
3107:
3098:
3089:
3084:
3080:
3072:
3071:
3070:
3056:
3053:
3047:
3044:
3041:
3035:
3012:
3004:
3000:
2982:
2978:
2969:
2960:
2952:
2948:
2930:
2926:
2917:
2908:
2903:
2899:
2891:
2890:
2889:
2872:
2869:
2866:
2860:
2852:
2848:
2843:
2841:
2822:
2817:
2813:
2809:
2806:
2801:
2797:
2793:
2790:
2787:
2780:
2779:
2778:
2777:
2758:
2755:
2752:
2729:
2704:
2700:
2696:
2691:
2687:
2675:
2667:
2665:
2663:
2659:
2656:-dimensional
2655:
2651:
2647:
2643:
2639:
2635:
2627:
2611:
2608:
2605:
2598:
2581:
2575:
2567:
2554:
2553:
2552:
2532:
2529:
2526:
2520:
2517:
2513:
2504:
2500:
2482:
2478:
2469:
2460:
2452:
2448:
2430:
2426:
2417:
2407:
2401:
2396:
2393:
2390:
2386:
2382:
2376:
2356:
2355:
2354:
2352:
2348:
2325:
2322:
2319:
2312:
2306:
2298:
2285:
2284:
2283:
2281:
2273:
2268:
2266:
2246:
2243:
2237:
2231:
2228:
2202:
2199:
2195:
2191:
2188:
2180:
2162:
2158:
2150:
2144:
2137:
2134:
2105:
2101:
2093:
2087:
2080:
2077:
2050:
2027:
2024:
2021:
2017:
2008:
2004:
1995:
1991:
1982:
1977:
1966:
1958:
1954:
1944:
1940:
1931:
1926:
1914:
1908:
1903:
1900:
1897:
1893:
1889:
1886:
1882:
1873:
1869:
1858:
1851:
1848:
1835:
1827:
1823:
1812:
1805:
1802:
1788:
1782:
1777:
1774:
1771:
1767:
1763:
1756:
1755:
1754:
1752:
1731:
1724:
1721:
1714:
1711:
1706:
1699:
1696:
1689:
1686:
1683:
1657:
1654:
1650:
1641:
1637:
1623:
1616:
1613:
1606:
1594:
1586:
1582:
1568:
1561:
1558:
1551:
1538:
1532:
1527:
1524:
1521:
1517:
1513:
1507:
1499:
1486:
1485:
1484:
1482:
1478:
1462:
1454:
1452:
1447:
1440:
1437:
1436:
1435:
1433:
1429:
1410:
1407:
1403:
1397:
1390:
1387:
1375:
1371:
1362:
1353:
1348:
1341:
1338:
1326:
1322:
1313:
1303:
1297:
1292:
1289:
1286:
1282:
1278:
1272:
1264:
1255:
1249:
1246:
1241:
1238:
1228:
1227:
1226:
1212:
1189:
1186:
1183:
1180:
1177:
1171:
1163:
1147:
1142:
1131:
1126:
1099:
1094:
1083:
1078:
1064:
1061:
1058:
1052:
1029:
1026:
1023:
1017:
997:
994:
991:
988:
985:
982:
979:
957:
953:
945:
927:
923:
915:
911:
906:
902:
901:
891:
887:
883:
879:
874:
867:
865:
863:
859:
855:
851:
847:
843:
838:
836:
832:
831:of the system
830:
826:
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
763:
753:
750:
745:
739:
738:
737:
736:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
594:
590:
586:
582:
579:
578:
574:
571:
569:
566:
564:
561:
559:
556:
552:
549:
548:
547:
544:
542:
539:
537:
534:
532:
529:
528:
524:
519:
518:
510:
507:
505:
502:
498:
495:
493:
490:
489:
488:
485:
483:
480:
478:
475:
473:
468:
465:
461:
458:
457:
453:
450:
447:
443:
439:
438:
434:
431:
429:
426:
424:
421:
419:
414:
412:
409:
407:
404:
402:
399:
398:
391:
390:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
350:
348:
347:
342:
339:
338:
333:
332:
326:
325:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
278:
276:
273:
271:
267:
265:
262:
259:
255:
251:
249:
246:
244:
241:
239:
236:
234:
231:
227:
224:
222:
219:
218:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
186:
185:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
138:
131:
130:
124:
121:
119:
116:
114:
111:
110:
107:
103:
84:
81:
71:
65:
51:
50:
47:
43:
39:
38:
33:
19:
8278:
8264:
8246:
8230:
8221:. Retrieved
8209:
8196:. Retrieved
8187:
8179:
8170:. Retrieved
8161:
8141:
8130:
8115:
8096:
8087:
8044:
8040:
8034:
8022:
8018:
8006:
8002:
7961:
7955:
7946:
7933:
7926:
7917:
7913:
7900:
7890:. New York:
7886:
7876:
7864:
7860:
7855:
7822:
7818:
7812:
7803:
7630:
7161:
7157:
6882:
6584:
6376:
6372:
6107:
6105:
5877:
5643:
5640:
5494:
5308:
5221:
5040:
4779:
4329:
4070:
3893:Consider an
3892:
3776:
3753:
3649:
3642:
3543:
3459:
3330:
3327:
3027:
2844:
2837:
2671:
2653:
2631:
2550:
2350:
2346:
2344:
2277:
2042:
1672:
1476:
1450:
1443:
1438:
1425:
1161:
898:phase space
897:
895:
839:
829:trajectories
822:
806:
800:
591: /
587: /
585:displacement
583: /
444: /
406:Displacement
344:
335:
329:Formulations
316:Virtual work
256: /
196:Acceleration
189:Fundamentals
8279:Gravitation
8267:. S. Chand.
8047:: 201–210.
7756:, advanced
7733:for linear
3562:commutators
3550:mixed state
2351:Liouvillian
882:phase space
880:systems in
825:phase-space
815:statistical
727:von Neumann
394:Core topics
18:Liouvillian
8316:Categories
8223:January 6,
8198:January 6,
8172:January 6,
8054:1611.03303
7920:: 342–349.
7869:p. 16
7795:References
3656:observable
662:d'Alembert
642:Maupertuis
605:Scientists
487:Rigid body
161:Kinematics
8229:Uses the
7847:0022-2488
7790:(r-RESPA)
7739:viscosity
7594:α
7555:ω
7546:≡
7515:ω
7510:
7491:α
7466:ω
7456:−
7443:ω
7438:
7412:α
7409:−
7360:α
7356:−
7347:ω
7341:≡
7332:ω
7310:ω
7305:
7276:ω
7271:
7245:α
7242:−
7179:γ
7174:≡
7171:α
7131:δ
7128:γ
7119:−
7108:Γ
7040:∏
7019:δ
7016:γ
7013:−
6935:∏
6923:Γ
6891:δ
6853:δ
6850:γ
6847:−
6789:δ
6781:γ
6778:−
6762:∂
6749:∂
6735:∂
6728:−
6712:∂
6699:∂
6685:∂
6559:δ
6538:∂
6530:˙
6514:∂
6492:∂
6484:˙
6468:∂
6341:γ
6338:−
6322:∂
6314:∂
6308:−
6295:˙
6259:∂
6251:∂
6235:˙
6191:γ
6161:γ
6158:−
6081:α
6066:γ
6034:ω
5960:ω
5852:ω
5848:
5832:ω
5826:−
5819:ω
5815:
5762:ω
5758:
5749:ω
5723:ω
5719:
5614:Γ
5608:ρ
5599:Γ
5564:Γ
5529:δ
5513:Γ
5506:ρ
5476:Γ
5408:∏
5338:∏
5326:Γ
5171:ω
5103:∑
5015:δ
4994:∂
4986:˙
4970:∂
4948:∂
4940:˙
4924:∂
4818:δ
4794:Γ
4752:δ
4723:∂
4715:˙
4699:∂
4648:δ
4619:∂
4611:˙
4595:∂
4526:Γ
4498:δ
4387:˙
4351:˙
4302:δ
4296:˙
4241:δ
4235:˙
4159:δ
4130:Γ
4124:ρ
4101:Γ
3995:∏
3987:Γ
3865:−
3859:∈
3824:ω
3736:⟩
3718:⟨
3712:ℏ
3700:−
3694:⟩
3688:⟨
3622:ρ
3607:ℏ
3586:∂
3581:ρ
3578:∂
3515:ω
3468:ω
3430:ω
3379:ω
3290:μ
3282:∂
3278:∂
3260:μ
3238:μ
3230:∂
3226:∂
3208:μ
3186:μ
3178:∂
3174:∂
3164:μ
3156:∂
3148:∂
3142:−
3134:μ
3126:∂
3122:∂
3112:μ
3104:∂
3096:∂
3069:, we get
3005:μ
2997:∂
2993:∂
2983:μ
2975:∂
2967:∂
2961:−
2953:μ
2945:∂
2941:∂
2931:μ
2923:∂
2915:∂
2818:μ
2807:∧
2802:μ
2788:ω
2759:ω
2705:μ
2692:μ
2606:ρ
2599:^
2573:∂
2568:ρ
2565:∂
2533:∙
2521:−
2497:∂
2493:∂
2475:∂
2467:∂
2461:−
2445:∂
2441:∂
2423:∂
2415:∂
2387:∑
2377:^
2326:ρ
2304:∂
2299:ρ
2296:∂
2247:˙
2232:˙
2192:ρ
2155:∂
2138:˙
2128:∂
2098:∂
2081:˙
2071:∂
2001:∂
1988:∂
1974:∂
1967:−
1951:∂
1937:∂
1923:∂
1894:∑
1890:ρ
1866:∂
1852:˙
1842:∂
1820:∂
1806:˙
1796:∂
1768:∑
1764:ρ
1725:˙
1715:ρ
1700:˙
1690:ρ
1684:ρ
1634:∂
1617:˙
1607:ρ
1601:∂
1579:∂
1562:˙
1552:ρ
1546:∂
1518:∑
1505:∂
1500:ρ
1497:∂
1475:obeys an
1463:ρ
1448:uses the
1391:˙
1368:∂
1363:ρ
1360:∂
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1314:ρ
1311:∂
1283:∑
1270:∂
1265:ρ
1262:∂
1242:ρ
1172:ρ
1053:ρ
1018:ρ
992:…
878:classical
707:Liouville
589:frequency
509:Vibration
226:potential
151:Continuum
146:Celestial
123:Textbooks
8139:(1979).
8095:(2007).
8079:53691877
7884:(1902).
7867:(1906),
7777:See also
6997:′
6974:′
6926:′
6628:′
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6400:′
6183:, where
5567:′
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4856:′
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4569:′
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4071:We want
3797:Examples
1205:in time
972:, where
846:friction
762:Category
687:Hamilton
672:Lagrange
667:Clairaut
632:Horrocks
593:velocity
563:Pendulum
551:reactive
523:Rotation
492:dynamics
442:Inertial
428:Friction
311:Velocity
286:Momentum
166:Kinetics
156:Dynamics
134:Branches
118:Timeline
8059:Bibcode
7827:Bibcode
7709:Remarks
4402:denote
3777:In the
2650:measure
803:physics
722:Koopman
682:Poisson
677:Laplace
622:Huygens
617:Galileo
462: (
401:Damping
254:Inertia
248:Impulse
221:kinetic
171:Statics
141:Applied
113:History
8286:
8149:
8103:
8077:
8011:p. 110
7968:
7845:
6075:
5878:where
4330:where
3754:where
3654:of an
2776:2-form
2043:where
1160:. The
760:
712:Appell
697:Cauchy
692:Jacobi
637:Halley
627:Newton
612:Kepler
464:linear
460:Motion
306:Torque
281:Moment
216:Energy
206:Couple
8218:(PDF)
8075:S2CID
8049:arXiv
8027:p. 16
7993:(PDF)
7938:(PDF)
7910:(PDF)
7760:, or
1749:is a
1432:Gibbs
912:with
717:Gibbs
702:Routh
657:Euler
296:Speed
291:Space
233:Force
8284:ISBN
8225:2014
8200:2014
8174:2014
8147:ISBN
8101:ISBN
7966:ISBN
7843:ISSN
7658:and
6087:0.25
5905:and
4446:and
4366:and
3785:for
2636:and
2120:and
942:and
823:the
817:and
301:Time
264:Mass
8067:doi
8045:502
7835:doi
7745:or
7507:sin
7435:cos
7302:sin
7268:cos
6108:not
6072:0.5
5845:sin
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801:In
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