Nekhoroshev theorem for the periodic Toda lattice

TL;DR

This paper proves that the periodic Toda lattice's Hamiltonian is strictly convex in significant regions of phase space, enabling the application of Nekhoroshev's theorem to demonstrate long-term stability of the system.

Contribution

It establishes the strict convexity of the Toda Hamiltonian in key regions, extending Nekhoroshev's stability results to this integrable system.

Findings

Hamiltonian is strictly convex in the interior of the positive quadrant

Nekhoroshev's theorem applies to almost all of the phase space

Long-term stability results for the periodic Toda lattice

Abstract

The periodic Toda lattice with $N$ sites is globally symplectomorphic to a two parameter family of $N-1$ coupled harmonic oscillators. The action variables fill out the whole positive quadrant of $\R^{N-1}$. We prove that in the interior of the positive quadrant as well as in a neighborhood of the origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's theorem applies on (almost) all parts of phase space.