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

Understand

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 $r>1$.