Complete Integrability for Hamiltonian Systems with a Cone Potential

TL;DR

This paper establishes conditions under which Hamiltonian systems with cone potentials are completely integrable, providing smooth first integrals and demonstrating robustness under perturbations.

Contribution

It introduces a broad class of completely integrable Hamiltonian systems with cone potentials and shows their integrability persists under small potential perturbations.

Findings

Constructed a large class of integrable systems with cone potentials.

Proved the smoothness and involution of asymptotic velocity functions.

Demonstrated stability of integrability under compactly supported perturbations.

Abstract

It is known that, if a point in $R^n$ is driven by a bounded below potential $V$, whose gradient is always in a closed convex cone which contains no lines, then the velocity has a finite limit as time goes to $+\infty$. The components of the asymptotic velocity, as functions of the initial data, are trivially constants of motion. We find sufficient conditions for these functions to be $C^k$ ($2\le k \le+\infty$) first integrals, independent and pairwise in involution. In this way we construct a large class of completely integrable systems. We can deal with very different asymptotic behaviours of the potential and we have persistence of the integrability under any small perturbation of the potential in an arbitrary compact set.