How the Jacobi metrics may be useful in nonholonomic mechanics withorwithoutconstraints?

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: Because I know some mechanical implications

1. What results of riemannian geometry via can be transferred into implications on the dynamics of mechancal systems by way of Jacobi metricsmetric?
2. Furthermore, but they are limited to unconstrained sistemsin 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.

2 edited tags
1

How the Jacobi metrics may be useful in nonholonomic mechanics?

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.

Question: Because I know some mechanical implications of riemannian geometry via Jacobi metrics, but they are limited to unconstrained sistems, I would know what are examples of the usefulness of Jacobi metrics in nonholonomic mechanics.

Obviously any feedback is welcome. Thank you.