Integrable magnetic geodesic flows on 2-surfaces

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...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Authors: Agapov, Sergei, Potashnikov, Alexey, Shubin, Vladislav
Format: Article
Language:English
Subjects:
Energy levels
Exact solutions
Hamiltonian functions
Hodographs
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2205.12499