Abstract:
The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential a smooth function on a two-dimensional torus. It is known that the existence of integrals of degrees 1 and 2 is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree n independent of the energy integral, then there exists an additional integral of degree 1 or 2. In the present paper this result is established for n=3 (which generalizes a theorem of Byalyi), and for n=4, 5, and 6 this is proved under some additional assumptions about the spectrum of the potential.
Citation:
N. V. Denisova, V. V. Kozlov, “Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space”, Sb. Math., 191:2 (2000), 189–208
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\by N.~V.~Denisova, V.~V.~Kozlov
\paper Polynomial integrals of reversible mechanical systems with a~two-dimensional torus as the~configuration space
\jour Sb. Math.
\yr 2000
\vol 191
\issue 2
\pages 189--208
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V. V. Kozlov, “Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees”, Izv. Math., 87:5 (2023), 972–986
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N. V. Denisova, “On Momentum-Polynomial Integrals of a Reversible Hamiltonian System of a Certain Form”, Proc. Steklov Inst. Math., 310 (2020), 131–136
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N. V. Denisova, V. V. Kozlov, D. V. Treschev, “Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space”, Izv. Math., 76:5 (2012), 907–921
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