The local instability of simple relativistic systems (Q1373582)

By a simple relativistic dynamical system we mean a manifold \(M\) with a metric \(g\) having Lorentz signature \((-,+,\cdots , +)\), and with motions of particles on \(M\) subject to a real potential \(V\). The classical Maupertuis-Jacobi principle states that trajectories with total energy \(E\) of this system are geodesics of the Jacobi metric \(g_E =2(E-V)g\) on the submanifold \(\{x\in M: V(x)\leq E\}\). In this interesting paper, the authors formulate a generalized Maupertuis-Jacobi principle that states that the trajectories of the system above are pregeodesics with respect to the metric \(2|E-V|g\). The generalization allows to develop a condition for local instability of geodesics in terms of the sectional curvature of \(M\). The authors demonstrate this method to describe the behavior of several relativistic Hamiltonian systems, namely three examples of the Friedmann-Robertson-Walker cosmological model and the motion of a charged particle in a uniform magnetic field with a linearly polarized gravitational wave. Illustrations of the maps of Gauss curvature for these systems are included.