Geodesic
Tensor invariants of geodesic, potential, and dissipative systems on tangent bundles of two-dimensional manifolds
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 501 (2021), pp. 89-94

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Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles of smooth two-dimensional manifolds are presented. The connection between the presence of these invariants and the full set of first integrals necessary for the integration of geodesic, potential, and dissipative systems is shown. The force fields introduced into the considered systems make them dissipative with dissipation of different signs and generalize previously considered force fields.
Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form. 
@article{DANMA_2021_501_a16,
     author = {M. V. Shamolin},
     title = {Tensor invariants of geodesic, potential, and dissipative systems on tangent bundles of two-dimensional manifolds},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {89--94},
     publisher = {mathdoc},
     volume = {501},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_501_a16/}
}
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M. V. Shamolin. Tensor invariants of geodesic, potential, and dissipative systems on tangent bundles of two-dimensional manifolds. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 501 (2021), pp. 89-94. http://geodesic.mathdoc.fr/item/DANMA_2021_501_a16/