This article is cited in 15 papers

Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space

N. V. Denisova^a, V. V. Kozlov^b, D. V. Treschev<sup>b

a</sup> M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
^b Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.

Keywords: irreducible integrals, systems with impacts, spectrum of a potential.

UDC: 517.9+531.01

MSC: 37J15, 70F05, 70H07

Received: 25.04.2012

DOI: 10.4213/im8001

 English version: Izvestiya: Mathematics, 2012, 76:5, 907–921

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