Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type

In this paper, we deal with the classical question of the existence of polynomials in momenta integrals for geodesic flows on the 2-torus. For the quasilinear system on the coefficients of the polynomial integral, we investigate the region (so-called elliptic region) where two of the eigenvalues are...

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Bibliographic Details
Published in:Nonlinearity 2011-12, Vol.24 (12), p.3541-3554
Main Authors: Bialy, Misha, Mironov, Andrey E
Format: Article
Language:English
Subjects:
Classification
Conjugates
Eigenvalues
Fluid dynamics
Fluid flow
Integrals
Nonlinearity
Polynomials
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Summary:In this paper, we deal with the classical question of the existence of polynomials in momenta integrals for geodesic flows on the 2-torus. For the quasilinear system on the coefficients of the polynomial integral, we investigate the region (so-called elliptic region) where two of the eigenvalues are complex conjugate. We show that for quartic integrals the other two eigenvalues are real and necessarily genuinely nonlinear. This observation, together with the property of the system to be rich (semi-Hamiltonian), enables us to classify elliptic regions completely. We prove that on these regions the integral is always reducible. 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 can be found only within the region of hyperbolicity of the quasilinear system.
ISSN:0951-7715
1361-6544
DOI:10.1088/0951-7715/24/12/010