Jacobi metric

Summary

The Jacobi metric of a mechanical system with fixed total energy $E$ is a Riemannian metric on part of the configuration space. This part is the region where the potential energy is less than $E$. It is the kinetic-energy metric multiplied by twice the difference between $E$ and the potential. By Maupertuis's principle, the paths of the motions of energy $E$ are exactly its geodesics. So it turns questions about dynamics into questions of Riemannian geometry.