Sorting: The Gauss Thermostat, the Toda Lattice and Double Bracket Equations,

Abstract

In this paper we consider certain equations that have gradient like behavior and which sort numbers in an analog fashion. Two kinds of equation that have been discussed earlier that achieve this are the Toda lattice equations and the double bracket equations. The Toda lattice equations are Hamiltonian and can be shown to be a special type of double bracket equation. The double bracket equations themselves are gradient (and hence the Toda lattice has a dual Hamiltonian/gradient form). Here we compare these systems to a system that arises from imposing a constant kinetic energy constraint on a one dimensional forced system. 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. This system is neither Hamiltonian nor gradient.