Geodesic
Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of two-dimensional manifolds
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 2, Tome 209 (2022), pp. 108-116

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In this paper, we construct tensor invariants (differential forms) of homogeneous dynamical systems on the tangent bundles of smooth two-dimensional manifolds. We establish the relationship between the presence of such invariants and the existence of complete sets of first integrals, which are necessary for integrating geodesic, potential, and dissipative systems. Due to force fields, systems considered are dissipative; they are generalizations of systems considered earlier.
Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form. 
@article{INTO_2022_209_a8,
     author = {M. V. Shamolin},
     title = {Some tensor invariants of geodesic, potential, and dissipative systems on the tangent 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 = {108--116},
     publisher = {mathdoc},
     volume = {209},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_209_a8/}
}
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M. V. Shamolin. Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of two-dimensional manifolds. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 2, Tome 209 (2022), pp. 108-116. http://geodesic.mathdoc.fr/item/INTO_2022_209_a8/