The Hamilton–Jacobi method for non-Hamiltonian systems
2015, Vol. 11, No. 2, pp. 279-286
Author(s): Vedenyapin V. V., Fimin N. N.
Author(s): 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.
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