Integrable geodesic flows on 2-torus: formal solutions and variational principle
Abstract
In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in [3] that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metric of Egorov type. We use this fact in order to prove that in the case of integrals of degree three and four the system is in fact equivalent to a single remarkable equation of order 3 and 4 respectively. Remarkably the equation for the case of degree four has variational meaning: it is Euler-Lagrange equation of a variational principle. Next we prove that this equation for has formal double periodic solutions as a series in a small parameter.
Cite
@article{arxiv.1401.2244,
title = {Integrable geodesic flows on 2-torus: formal solutions and variational principle},
author = {Michael and Bialy and Andrey E. Mironov},
journal= {arXiv preprint arXiv:1401.2244},
year = {2014}
}