On Maupertuis principle in dynamics

A new statement of Maupertuis principle of extremal action is contributed on the basis of a constrained action principle in the velocity phase-space in which the condition of energy conservation is imposed on virtual velocities. Dynamical systems governed by time-dependent Lagrangians on nonlinear configuration manifolds and subject to the action of time-dependent forces are considered. In time-independent systems, and in particular in conservative systems, the constrained action principle specializes to a formulation of the original Maupertuis least action principle in which however conservation of energy along the trajectory is a natural consequence of the variational principle and not an a priori assumption as in classical statements