Curvatures and hyperbolic flows for natural mechanical systems in Finsler geometry

Abstract

We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (involving the gradient of the potential) of the Finsler manifold and the Hessian of the potential w.r.t. a Riemannian metric induced from the Finslerian metric. As an application, we give sufficient conditions for the Hamiltonian flows of the least action to be hyperbolic and show new examples of Anosov flows.