Nos\'e-Thermostated Mechanical Systems on the n-Torus

TL;DR

This paper demonstrates the existence of positive-measure invariant tori in Nosé-thermostated mechanical systems on the n-torus, especially near the infinite temperature limit, extending previous results to variable mass thermostats.

Contribution

It proves the presence of invariant tori in Nosé-thermostated systems on the n-torus, including variable mass thermostats, near the infinite temperature limit.

Findings

Positive-measure invariant tori exist near infinite temperature.

Results apply to all thermostats similar to Nosé's.

Extends previous work to variable mass thermostats.

Abstract

Let $H(q,p) = \frac12 | p |^2 + V(q)$ be an $n$-degree of freedom $C^r$ mechanical Hamiltonian on the cotangent bundle of the $n$-torus where $r>2n+2$. When the metric $| * |$ is flat, the Nos\'e-thermostated system associated to $H$ is shown to have a positive-measure set of invariant tori near the infinite temperature limit. This is shown to be true for all variable mass thermostats similar to Nos\'e's, too. These results complement results of Legoll, Luskin & Moeckel and the author.