Nonholonomic double-bracket equations and the Gauss thermostat

In this Rapid Communication we consider certain equations that arise from imposing a constant kinetic-energy constraint on a one-dimensional set of oscillators. This is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with Gauss's law of least constraint. Dynamics of this sort are of interest in nonequilibrium molecular dynamics. We show that under certain choices of external potential these equations give rise to a generalization of the so-called double-bracket equations which are of interest in studying gradient flows and integrable systems such as the Toda lattice. In the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained.