The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems
2018, Vol. 14, no. 4, pp. 495-501
Author(s): Fimin N. N., Chechetkin V. M.
This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License
Author(s): Fimin N. N., Chechetkin V. M.
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.
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