Valery Chechetkin

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 N-linear 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.

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.