How the Jacobi metrics may be useful in mechanics with or without constraints?

Question

A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$

If $V{<}e$ then $g_J=(e-V)g$ is the Jacobi metric on $Q,$ and Maupertuis' principle states that, up to reparametrization, the trajectories of motion of $(Q,K,V)$ with energy $e$ are exactly the geodesics of $g_J.$ (which by the way are the trajectory of motion of $(Q,K_{g_J},0).$

My question is motivated by what I read in Jair Koiller's paper "Reduction of Some Classical Non-Holonomic Systems with Symmetry", Arch. Rational Mech. Anal. 118 (1992).

The Jacobi metric is useful for constrained systems as well, an observation which seems not to have been sufficiently explored in the theory of nonholonomic systems.

Koiller goes on saying that, if $D,$ a tangent distribution on $Q,$ represents a linear constraint on the velocities, then the trajectories of motion of the constrained system $(Q,K,V;D)$ with energy $e$ are the same as the ones of $(Q,K_{g_J},0;D),$ with energy $1,$ up to a reparametrization.

Edited Question:

1. What results of riemannian geometry can be transferred into implications on the dynamics of mechancal systems by way of Jacobi metric?
2. Furthermore, in the light of the quotation, I would know: what are examples of the usefulness of Jacobi metrics in nonholonomic mechanics?

Obviously any feedback is welcome. Thank you.

Answer 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 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.