Integrable geodesic flows on 2-torus: formal solutions and variational principle

TL;DR

This paper investigates polynomial first integrals of geodesic flows on a 2-torus, showing the associated system is semi-Hamiltonian and of Egorov type, with variational principles and formal solutions for degrees three and four.

Contribution

It establishes the Egorov property of the metric, reduces the system to single equations for degrees three and four, and demonstrates the variational nature and formal solutions for degree four.

Findings

The metric is of Egorov type.

System reduces to a single equation for degrees 3 and 4.

Degree four equation has formal double periodic solutions.

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 $n=4$ has formal double periodic solutions as a series in a small parameter.