Cubic and Quartic integrals for geodesic flow on 2-torus via system of Hydrodynamic type

2011/01/18 by Michael Michael, Michael, Bialy +2
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #math.DG

paper · pdf · doi:10.48550/arxiv.1101.3449

corrected typos

openalex publication_date 2011/01/18 · arxiv created 2013/03/29 · arxiv updated 2013/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of the polynomial integral we consider the region (so called elliptic regions) where there are complex-conjugate eigenvalues. We show that for quartic integrals in the other two eigenvalues are real and genuinely nonlinear. This observation together with the property of the system to be Rich (Semi-Hamiltonian) enables us to classify elliptic regions completely. The case of complex-conjugate eigenvalues for the system corresponding to the integral of degree 3 is done similarly. These results show that if new integrable examples exist they could be found only within the region of Hyperbolicity of the quasi-linear system.

Citations

• Richness or Semi-Hamiltonicity of quasi-linear systems which are not in evolution form

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