On first integrals of geodesic flows on a two-torus

TL;DR

This paper investigates the existence of polynomial first integrals in geodesic flows on a two-torus, linking the problem to soliton equations and proving nonexistence results for quadratic integrals in specific cases.

Contribution

It establishes the nonexistence of quadratic first integrals for certain magnetic geodesic flows and connects the problem to stationary dispersionless soliton equations.

Findings

No quadratic first integrals exist for certain magnetic geodesic flows.

The problem relates to stationary dispersionless limits of soliton equations.

Provides conditions under which additional polynomial integrals do not exist.

Abstract

The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.