Tensor invariants of geodesic, potential and dissipative systems. IV. Systems on tangents bundles of $n$-dimensional manifolds

In this paper, we present tensor invariants (first integrals and differential forms) for dynamical systems on the tangent bundles of smooth $n$-dimensional manifolds separately for $n=1$, $n=2$, $n=3$, $n=4$, and for any finite $n$. We demonstrate the connection between the existence of these invariants and the presence of a full set of first integrals that are necessary for integrating geodesic, potential, and dissipative systems. The force fields acting in systems considered make them dissipative (with alternating dissipation).
The first part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 227 (2023), pp. 100–128. The second part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 228 (2023), pp. 92–118. The third part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 229 (2023), pp. 90–119.