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Nelin. Dinam., 2015, Volume 11, Number 2, Pages 279–286 (Mi nd480)  

This article is cited in 6 scientific papers (total in 6 papers)

Original papers

The Hamilton – Jacobi method for non-Hamiltonian systems

V. V. Vedenyapin, N. N. Fimin

Keldysh Institute of Applied Mathematics Miusskaya sq. 4, Moscow, 125047, Russia

Abstract: The hydrodynamic substitution applied earlier only in the theory of plasma represents the decomposition of a special type of the distribution function in phase space which is marking out obviously dependence of a momentum variable on a configuration variable and time. For the system of the autonomous ordinary differential equations (ODE) given to a Hamilton form, evolution of this dynamic system is described by the classical Liouville equation for the distribution function defined on the cotangent bundle of configuration manifold. Liouvilles equation is given to the reduced Eulers system representing pair of uncoupled hydrodynamic equations (continuity and momenta transfer). The equation for momenta by simple transformations can bebrought to the classicalequation of Hamilton – Jacobi foreikonal function. For the general systemautonomous ODE it is possibleto enter the decomposition of configuration variables into new configuration and «momenta» variables. In constructed thus phase (generally speaking, asymmetrical) space it is possible to consider the generalized Liouvilles equation, to lead it again to the pair of the hydrodynamic equations. The equation of transfer of «momenta is an analog of the Hamilton – Jacobi equation for the general non-Hamilton case.

Keywords: hydrodynamical substitution, Liouville equation, Hamilton – Jacobi method, non-Hamiltonian system

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00670

Full text: PDF file (322 kB)
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UDC: 517.9
MSC: 34A25
Received: 27.11.2014
Revised: 24.02.2015

Citation: V. V. Vedenyapin, N. N. Fimin, “The Hamilton – Jacobi method for non-Hamiltonian systems”, Nelin. Dinam., 11:2 (2015), 279–286

Citation in format AMSBIB
\by V.~V.~Vedenyapin, N.~N.~Fimin
\paper The Hamilton\,--\,Jacobi method for non-Hamiltonian systems
\jour Nelin. Dinam.
\yr 2015
\vol 11
\issue 2
\pages 279--286

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    This publication is cited in the following articles:
    1. V. V. Vedenyapin, M. A. Negmatov, N. N. Fimin, “Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences”, Izv. Math., 81:3 (2017), 505–541  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. N. N. Fimin, V. M. Chechetkin, “Cistemy kvazilineinykh uravnenii s odinakovoi glavnoi chastyu i gidrodinamicheskaya podstanovka”, Preprinty IPM im. M. V. Keldysha, 2018, 055, 12 pp.  mathnet  crossref  elib
    3. N. N. Fimin, V. M. Chechetkin, “Primenenie gidrodinamicheskoi podstanovki dlya sistem uravnenii s odinakovoi glavnoi chastyu”, Nelineinaya dinam., 14:1 (2018), 53–61  mathnet  crossref  elib
    4. V. V. Vedenyapin, S. Z. Adzhiev, V. V. Kazantseva, “Entropiya po Boltsmanu i Puankare, ekstremali Boltsmana i metod Gamiltona–Yakobi v negamiltonovoi situatsii”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 37–59  mathnet  crossref
    5. V. V. Vedenyapin, “Uravnenie Vlasova–Maksvella–Einshteina”, Preprinty IPM im. M. V. Keldysha, 2018, 188, 20 pp.  mathnet  crossref  elib
    6. V. V. Vedenyapin, I. S. Pershin, “Vlasov–Maxwell–Einstein equation and Einstein lambda”, Preprinty IPM im. M. V. Keldysha, 2019, 039, 17 pp.  mathnet  crossref
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