Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of two-dimensional manifolds

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.