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arXiv · 2411.18920

Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method

Sergei Agapov
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Abstract

We study Riemannian metrics on 2-surfaces with integrable geodesic flows such that an additional first integral is high-degree polynomial in momenta. This problem reduces to searching for solutions to certain quasi-linear systems of PDEs which turn out to be semi-Hamiltonian. We construct plenty of local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5. Our construction is essentially based on applying the generalized hodograph method.

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Sergei Agapov. 2024-11-28. Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method. https://doi.org/10.1016/j.geomphys.2025.105629

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