Invariant volume forms of geodesic, potential, and dissipative systems on a tangent bundle of a four-dimensional manifold

Complete sets of invariant differential forms of phase volume for homogeneous dynamical systems on tangent bundles of smooth four-dimensional manifolds are presented. The connection between the existence of these invariants and the complete set of first integrals necessary for the integration of geodesic, potential, and dissipative systems is shown. The introduced force fields make the considered systems dissipative with dissipation of different signs and generalize previously considered fields.