Geodesic
Tensor invariants of geodesic, potential and dissipative systems. I. Systems on tangents bundles of two-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 1, Tome 227 (2023), pp. 100-128

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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).
Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form 
@article{INTO_2023_227_a7,
     author = {M. V. Shamolin},
     title = {Tensor invariants of geodesic, potential and dissipative systems. {I.} {Systems} on tangents bundles of two-dimensional manifolds},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {100--128},
     publisher = {mathdoc},
     volume = {227},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_227_a7/}
}
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M. V. Shamolin. Tensor invariants of geodesic, potential and dissipative systems. I. Systems on tangents bundles of two-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 1, Tome 227 (2023), pp. 100-128. http://geodesic.mathdoc.fr/item/INTO_2023_227_a7/