Nikolay Fimin
Miusskaya pl. 4, Moscow, 125047, Russia
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
Publications:
Fimin N. N., Chechetkin V. M.
The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems
2018, Vol. 14, no. 4, pp. 495501
Abstract
Geometrization of the description of vortex hydrodynamic systems can be made on the basis
of the introduction of the Monge – Clebsch potentials, which leads to the Hamiltonian form
of the original Euler equations. For this, we construct the kinetic Lagrange potential with the
help of the flow velocity field, which is preliminarily determined through a set of scalar Monge
potentials, and thermodynamic relations. The next step is to transform the resulting Lagrangian
by means of the Legendre transformation to the Hamiltonian function and correctly introduce
the generalized impulses canonically conjugate to the configuration variables in the new phase
space of the dynamical system. Next, using the Hamiltonian function obtained, we define the
Hamiltonian space on the cotangent bundle over the Monge potential manifold. Calculating the
Hessian of the Hamiltonian, we obtain the coefficients of the fundamental tensor of the Hamiltonian
space defining its metric. Next, we determine analogs of the Christoffel coefficients for
the Nlinear connection. Considering the Euler – Lagrange equations with the connectivity coefficients
obtained, we arrive at the geodesic equations in the form of horizontal and vertical paths
in the Hamiltonian space. In the case under study, nontrivial solutions can have only differential
equations for vertical paths. Analyzing the resulting system of equations of geodesic motion
from the point of view of the stability of solutions, it is possible to obtain important physical
conclusions regarding the initial hydrodynamic system. To do this, we investigate a possible
increase or decrease in the infinitesimal distance between the geodesic vertical paths (solutions
of the corresponding system of Jacobi – Cartan equations). As a result, we can formulate very
general criterions for the decay and collapse of a vortex continual system.

Fimin N. N., Chechetkin V. M.
Application of the hydrodynamic substitution for systems of equations with the same principal part
2018, Vol. 14, no. 1, pp. 5361
Abstract
The properties of quasilinear 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 nonHamiltonian) firstorder autonomous differential equations. Examples are given of the use of the developed formalism for systems of gravitating particles in postNewtonian 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.

Vedenyapin V. V., Fimin N. N.
The Hamilton–Jacobi method for nonHamiltonian systems
2015, Vol. 11, No. 2, pp. 279286
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. 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 nonHamilton case.
