Integrable magnetic geodesic flows on 2-surfaces ^*

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 partial differential equations (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.

New examples of non-polynomial integrals of two-dimensional geodesic flows *

“…Various properties of such systems were also discussed in [12]. A similar situation also takes place for integrable magnetic geodesic flows with additional first integrals at a fixed energy level [13][14][15][16].…”

“…We reduce this problem to a certain quasi-linear system of PDEs and apply the generalized hodograph method to construct its exact solutions. Recently the same approach has yielded new integrable examples in [16,29] in slightly different situations.…”