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 Nelin. Dinam., 2018, Volume 14, Number 4, Pages 495–501 (Mi nd627)

Nonlinear physics and mechanics

The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems

N. N. Fimin, V. M. Chechetkin

Keldysh Institute of Applied Mathematics of RAS, Miusskaya pl. 4, Moscow, 125947 Russia

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 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.

Keywords: vortex dynamics, geodesic deviation, Monge manifold, metric tensor

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations 28 Russian Foundation for Basic Research 16-02-00656-A The work was supported by the Program of the Presidium of the Russian Academy of Sciences No. 28 “Cosmos: Fundamental Research Processes and their Interrelationships” and the RFFI Grant No. 16-02-00656-A.

DOI: https://doi.org/10.20537/nd180405

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MSC: 76B47
Accepted:14.11.2018
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Citation: N. N. Fimin, V. M. Chechetkin, “The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems”, Nelin. Dinam., 14:4 (2018), 495–501

Citation in format AMSBIB
\Bibitem{FimChe18} \by N. N. Fimin, V. M. Chechetkin \paper The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems \jour Nelin. Dinam. \yr 2018 \vol 14 \issue 4 \pages 495--501 \mathnet{http://mi.mathnet.ru/nd627} \crossref{https://doi.org/10.20537/nd180405} \elib{https://elibrary.ru/item.asp?id=36686070}