Killing equations in classical mechanics

The relation between Noether theorem and the generalized symmetries of Riemannian and Finsler metrics is explored to set in a more general framework the search of conserved quantities in classical mechanics. The Killing-tensor equations are written in the Jacobi metric, which constitutes the simplest way to ''geometrize'' the evolution of a conservative Lagrangian system: the relation between the additional invariant in the physical gauge and the Killing-tensor invariant on the Jacobi manifold is written in the general case and is explicited in several classical cases, for which all the Killing tensors associated to quadratic invariants at arbitrary energy are found. This approach allows to intel pl et the Killing tensors as the generators of generalized Noetherian transformations depending also on the velocities. After that, the Killing-vector equations are written in the Finsler metric, which allows a more general geometrization of dynamical problems, including non-conservative systems and Lagrangian with terms linear in the velocities. In fact, exploiting the velocity dependence of tensorial objects in the Finsler metric, Killing vectors in this case give also the invariants of higher ol der in the momenta.