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Invariant Forms of Geodesic, Potential, and Dissipative Systems on Tangent Bundles of Finite-Dimensional Manifolds

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Abstract

It is well known [1–3] that a system of differential equations can be exactly integrated if a sufficient number of its tensor invariants (not only first integrals) are found. For example, if there is an invariant differential form of the phase volume, the number of required first integrals can be reduced. For conservative systems, this fact is natural, but for systems with attracting or repelling limit sets, not only some of the first integrals, but also the coefficients of available invariant differential forms should, generally speaking, include functions with essential singularities (see also [4–6]). In this paper, for the class of dynamical systems under consideration, we present complete sets of invariant differential forms for homogeneous systems on tangent bundles of smooth finite-dimensional manifolds.

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  1. Lomonosov Moscow State University, Moscow, Russia

    M. V. Shamolin

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  1. M. V. Shamolin
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Correspondence to M. V. Shamolin.

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Translated by I. Ruzanova

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Shamolin, M.V. Invariant Forms of Geodesic, Potential, and Dissipative Systems on Tangent Bundles of Finite-Dimensional Manifolds. Dokl. Math. 108, 248–255 (2023). https://doi.org/10.1134/S1064562423700941

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  • DOI: https://doi.org/10.1134/S1064562423700941

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