On the open Toda chain with external forcing

TL;DR

This paper analyzes the open Toda chain with external forcing, establishing its long-term behavior, integrability, and action-angle variables when the forcing stretches the system, while leaving the compressing case open for future work.

Contribution

It demonstrates the complete integrability of the forced open Toda chain with stretching forcing and constructs action-angle variables, extending the understanding of such systems.

Findings

System is completely integrable with N integrals

Constructed action-angle variables for the flow

Longtime behavior analyzed for stretching forcing

Abstract

We consider the open Toda chain with external forcing, and in the case when the forcing stretches the system, we derive the longtime behavior of solutions of the chain. Using an observation of J\"{u}rgen Moser, we then show that the system is completely integrable, in the sense that the $2N$-dimensional system has $N$ functionally independent Poisson commuting integrals, and also has a Lax-Pair formulation. In addition, we construct action-angle variables for the flow. In the case when the forcing compresses the system, the analysis of the flow remains open.