A class of integrable Hamiltonian systems including scattering of particles on the line with repulsive interactions

TL;DR

This paper introduces a new class of integrable Hamiltonian scattering systems with cone potentials, including particle interactions like inverse power and Calogero systems, and demonstrates their stability under perturbations.

Contribution

It develops a general integrability framework for cone potential systems, covering various particle interactions and normal form reductions, including known models like Calogero and Toda.

Findings

The systems can be integrated via asymptotic velocity.

A global canonical diffeomorphism to normal form is established.

Integrability persists under small perturbations.

Abstract

The main purpose of this paper is to introduce a new class of Hamiltonian scattering systems of the cone potential type that can be integrated via the asymptotic velocity. For a large subclass, the asymptotic data of the trajectories define a global canonical diffeomorphism $A$ that brings the system into the normal form $\dot P=0$, $\dot Q=P$. The integrability theory applies for example to a system of $n$ particles on the line interacting pairwise through rather general repulsive potentials. The inverse $r$-power potential for arbitrary $r>0$ is included, the reduction to normal form being carried out for the exponents $r>1$. In particular, the Calogero system is obtained for $r=2$. The treatment covers also the nonperiodic Toda lattice. The cone potentials that we allow can undergo small perturbations in any arbitrary compact set without losing the integrability and the reduction to normal form.