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    Nikolay Fimin

    Miusskaya pl. 4, Moscow, 125047, Russia
    Keldysh Institute of Applied Mathematics, Russian Academy of Sciences


    Fimin N. N., Chechetkin V. M.
    The properties of quasi-linear differential equations with the same the principal part are considered. Their connection with the reduced system of Euler equations is established, which results from the hydrodynamic substitution in the kinetic Liouville and Vlasov equations. When considering the momentum equation of the Euler system, it turns out that it reduces to a special form such as Liouville – Jacobi equation. This equation can also be investigated using a hydrodynamic substitution, but of conjugate type. The application of this substitution (of the second order) makes it possible to symmetrize the technique of applying hydrodynamic substitution and to extend the class of equations of hydrodynamic type to which systems of (in the general case non-Hamiltonian) first-order autonomous differential equations. Examples are given of the use of the developed formalism for systems of gravitating particles in post-Newtonian approximation and the hydrodynamic systems described by Monge potentials, with the aim of constructing the Liouville – Jacobi equations and applying to them a modified hydrodynamic substitution.
    Keywords: Liouville equation, quasi-linear equations, hydrodynamic substitution, Monge potentials, equations with the same principal part
    Citation: Fimin N. N., Chechetkin V. M.,  Application of the hydrodynamic substitution for systems of equations with the same principal part, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  53-61
    Vedenyapin V. V., Fimin N. N.
    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. Liouville’s equation is given to the reduced Euler’s system representing pair of uncoupled hydrodynamic equations (continuity and momenta transfer). The equation for momenta by simple transformations can be brought to the classical equation of Hamilton–Jacobi for eikonal function. For the general system autonomous ODE it is possible to 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 Liouville’s 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
    Citation: Vedenyapin V. V., Fimin N. N.,  The Hamilton–Jacobi method for non-Hamiltonian systems, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp.  279-286

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