Invariants of Dynamical Systems with Dissipation on Tangent Bundles of Low-Dimensional Manifolds

Abstract

Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles to smooth two-dimensional manifolds are presented in this paper. The connection between the presence of these invariants and the full set of the first integrals necessary for the integration of geodesic, potential and dissipative systems is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones. We also represent the typical examples from rigid body dynamics.