Geodesic
Tensor invariants of geodesic, potential and dissipative systems. III. Systems on tangents bundles of four-dimensional manifolds
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 3, Tome 229 (2023), pp. 90-119

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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). The first part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 227 (2023), pp. 100–128. The second part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 228 (2023), pp. 92–118.
Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form 
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     author = {M. V. Shamolin},
     title = {Tensor invariants of geodesic, potential and dissipative systems. {III.} {Systems} on tangents bundles of four-dimensional manifolds},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {90--119},
     publisher = {mathdoc},
     volume = {229},
     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2023_229_a8/}
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M. V. Shamolin. Tensor invariants of geodesic, potential and dissipative systems. III. Systems on tangents bundles of four-dimensional manifolds. 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 3, Tome 229 (2023), pp. 90-119. http://geodesic.mathdoc.fr/item/INTO_2023_229_a8/