Integrable magnetic geodesic flows on 2-torus: new example via quasi-linear system of PDEs

TL;DR

This paper constructs a new example of integrable magnetic geodesic flow on a 2-torus by deforming a Liouville metric with zero magnetic field into one with a small magnetic field, solving a semi-hamiltonian PDE system.

Contribution

It introduces a novel method to generate integrable magnetic geodesic flows on the 2-torus via deformation of metrics, providing explicit solutions to a semi-hamiltonian PDE system.

Findings

Existence of smooth periodic solutions to the semi-hamiltonian PDE system.

Deformation of Liouville metrics yields integrable magnetic flows.

New example of magnetic geodesic flow with quadratic integral.

Abstract

The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on one energy level is considered. This problem can be reduced to a remarkable Semi-hamiltonian system of quasi-linear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville metric with zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of a quadratic in momenta integral. Thus our construction gives a new example of smooth periodic solution to the Semi-hamiltonian (Rich) quasi-linear system of PDEs.