Maupertuis's principle

Remarks

The functional $S_0$ is the length functional of the Jacobi metric $2(E - V)\,g$ on $M_E$. So the paths of motion of energy $E$ are the geodesics of the Jacobi metric. This metric is a conformal change of the kinetic metric $g$ of the Riemannian manifold $(M, g)$. Along a true motion $\norm{\dot q}_g = \sqrt{2(E - V)}$, so $S_0$ equals $\int 2T\,dt = \int p \cdot dq$, the classical form of the abbreviated action. The proof compares the Euler–Lagrange equations of the length of the Jacobi metric with those of $L$, after the change of time $dt = \norm{\gamma'}_g / \sqrt{2(E - V)}\, ds$. Unlike Hamilton's principle, the variation fixes the energy and leaves the travel time free. For a free particle ($V = 0$), the paths are the geodesics of $g$ itself. The hypothesis $V < E$ is needed: where $V = E$ the Jacobi metric degenerates, and these are the turning points of the motion.

Going deeper

For a student

Maupertuis's principle turns a dynamics problem into a geometry problem. Orbits of energy $E$ become geodesics of the Jacobi metric $2(E - V)g$, and tools of Riemannian geometry, such as curvature and comparison theorems, then give information on the stability of orbits. It is derived from the stationary-action principle by fixing the energy instead of the time.