Andrey Mironov

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

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. We consider the problem of integrability on one energy level. 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 quasilinear system of PDEs. The talk is based on the joint paper with Misha Bialy (Tel-Aviv) and Sergey Agapov (Novosibirsk).

XXXVI Workshop on Geometric Methods in Physics	2-8.07.2017
VI School on Geometry and Physics	26-30.06.2017

Event sponsored by:
		University of Bialystok