Optical mechanical analogy and nonlinear nonholonomic constraints

In this paper we establish a connection between particle trajectories subject to a nonholonomic constraint and light ray trajectories in a variable index of refraction. In particular, we extend the analysis of systems with linear nonholonomic constraints to the dynamics of particles in a potential subject to nonlinear velocity constraints. We contrast the long time behavior of particles subject to a constant kinetic energy constraint (a thermostat) to particles with the constraint of parallel velocities. We show that, while in the former case the velocities of each particle equalize in the limit, in the latter case all the kinetic energies of each particle remain the same.