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Dear Giuseppe,

I think that the idea is that much more is known about the dynamics of geodesic flows in Riemannian and Finsler manifolds than about the dynamics of more general Hamiltonian systems. Here are two examples where the Jacobi metric is or has been useful:

1. A generalized spherical pendulum: If $V$ is a smooth potential function on the two-sphere and $e > \max V$, then there are infinitely many periodic orbits at the energy level $e$.

This is because of the Franks-Bangert-Hingston result stating that any Riemannian metric on the two-sphere has infinitely many prime closed geodesics.

2. Every smooth, compact, convex hypersurface in $({\mathbb R}^{2n}, \omega_{can)}$ carries a closed characteristic.

This is a celebrated theorem of A. Weinstein that has been widely generalized, but the original proof reduced the problem to finding a closed geodesic for non-reversible Finsler metrics on spheres, which you can do using Birkhoff's minimax procedure.

I'm less familiar with the constrained case, but I guess the answer may change if you consider non-holonomic systems (which are not Hamiltonian systems) or vakonomic systems.

1

Dear Giuseppe,

I think that the idea is that much more is known about the dynamics of geodesic flows in Riemannian and Finsler manifolds than about the dynamics of more general Hamiltonian systems. Here are two examples where the Jacobi metric is or has been useful:

1. A generalized spherical pendulum: If $V$ is a smooth potential function on the two-sphere and $e > \max V$, then there are infinitely many periodic orbits at the energy level $e$.

This is because of the Franks-Bangert-Hingston result stating that any Riemannian metric on the two-sphere has infinitely many prime geodesics.

2. Every smooth compact hypersurface in $({\mathbb R}^{2n}, \omega_{can)}$ carries a closed characteristic.

This is a celebrated theorem of A. Weinstein that has been widely generalized, but the original proof reduced the problem to finding a closed geodesic for non-reversible Finsler metrics on spheres, which you can do using Birkhoff's minimax procedure.

I'm less familiar with the constrained case, but I guess the answer may change if you consider non-holonomic systems (which are not Hamiltonian systems) or vakonomic systems.