Tensor invariants of geodesic, potential and dissipative systems. I. Systems on tangents bundles of two-dimensional manifolds

M. V. Shamolin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international winter mathematical school "Modern methods of function theory and related problems", Voronezh, January 27 - February 1, 2023, Part 1, Tome 227 (2023), pp. 100-128.

Résumé

In this paper, we present tensor invariants (first integrals and differential forms) for dynamical systems on the tangent bundles of smooth $n$-dimensional manifolds separately for $n=1$, $n=2$, $n=3$, $n=4$, and for any finite $n$. We demonstrate the connection between the existence of these invariants and the presence of a full set of first integrals that are necessary for integrating geodesic, potential, and dissipative systems. The force fields acting in systems considered make them dissipative (with alternating dissipation).

Détail

Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form

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