Invariant forms of geodesic, potential, and dissipative systems on tangent bundles of finite-dimensional manifolds

As is well-known [1–3], finding a sufficient number of tensor invariants (not only the first integrals) allows you to accurately integrate a system of differential equations. For example, the presence of an invariant differential form of the phase volume makes it possible to reduce the number of required first integrals. For conservative systems, this fact is natural, but for systems with attractive or repulsive limit sets, not only some first integrals, but also the coefficients of the available invariant differential forms should, generally speaking, include functions with essentially special points (see also [4–6]). In this paper, complete sets of invariant differential forms for homogeneous systems on tangent bundles to smooth finite-dimensional manifolds are presented for the class of dynamical systems under consideration.