Gyroscopically Stabilized Oscillators and Heat Baths

Abstract

In this paper we analyze the stability of a gyroscopic oscillator interacting with a finite- and infinite-dimensional heat bath in both the classical and quantum cases. We consider a finite gyroscopic oscillator model of a particle on a rotating disc and a particle in a magnetic field and we examine stability before and after coupling to a heat bath. The heat bath is modelled in the finite-dimensional setting by a system of independent oscillators with mass. It is shown that if the oscillator is gyroscopically stable, coupling to a sufficiently massive heat bath induces instability even in the finite-dimensional setting. The key mechanism for instability in this paper is thus not induced by damping. The meaning of these ideas in the quantum context is discussed. The model extends the exact diagonalization analysis of an oscillator and field of Ford, Lewis, and O'Connell to the gyroscopic setting. We also discuss the interesting role that damping of Landau type plays in the infinite limit.