The generalized Maupertuis principle (Q1286787)

For a Lagrangian of type \(L=T-V\), where the kinetic energy function \(T\) comes from a Riemannian metric \(g\), the classical Maupertuis principle states that trajectories with total energy \(E\) are geodesics of the Jacobi metric \(g_E = 2(E-V)g\) (at least on the ``admissible configuration space'' determined by \(V<E\)). Here the author discusses the generalized situation where the metric can have e.g. Lorentzian signature, and the Jacobi metric is taken as \(g_E = 2|E-V|g\). He proves existence and uniqueness of geodesics of \(g_E\) passing through the singular boundary where \(|E-V|=0\). The main theorem states that trajectories of the system with Lagrangian \(L\) are pregeodesics of \(g_E\), carrying a number of isolated points on the singular boundary or a number of line segments and isolated points. This extension of the Maupertuis principle is relevant for applications in general relativity.