The generalized Maupertuis principle

It is proved that trajectories of any mechanical system $(M,g,V)$, where $M$ is the configuration space, $g$ the metric defined by the kinetic energy form, and $V$ a potential function, with the natural Lagrangian, are pregeodesics with respect to the Jacobi metric $g_E=2|E-V|g$, $E$ being the total energy of the system.