Integrable magnetic geodesic flows on 2-surfaces

TL;DR

This paper investigates magnetic geodesic flows on 2-surfaces with special integrals, deriving exact solutions to associated PDEs using advanced mathematical methods, thus advancing understanding of integrable systems in differential geometry.

Contribution

It constructs explicit solutions for semi-Hamiltonian PDE systems arising from magnetic geodesic flows with quadratic or rational integrals, using hodograph, Legendre, and separation techniques.

Findings

Derived exact local solutions for semi-Hamiltonian PDEs

Applied generalized hodograph and Legendre methods

Enhanced understanding of integrable magnetic geodesic flows

Abstract

We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta integral, and also the case of a rational in momenta integral with a linear numerator and denominator. In both cases certain semi-Hamiltonian systems of PDEs appear. In this paper we construct exact solutions (generally speaking, local ones) to these systems: in the first case via the generalized hodograph method, in the second case via the Legendre transformation and the method of separation of variables.