The power of the Maupertuis-Jacobi principle—Dreams and reality

In this paper we examine the old idea of viewing the Hamiltonian systems as geodesic flows. This elegant geometrical description was expected to provide a starting point for application of ergodic concepts in stellar dynamics. We discuss the range of applicability of this approach. Perspectives for the future work concerning integrability and thermodynamical description of Hamiltonian systems are pointed out.