Abstract
We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an “irreducible” integral of the geodesic flow on the torus does not exceed 2.
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References
G. D. Birkhoff,Dynamical Systems, Amer. Math. Soc., New York (1927).
V. V. Kozlov and N. V. Denisova, “Polynomial integrals of geodesic flows on the two-dimensional torus,”Mat. Sb. [Russian Acad. Sci. Sb. Math.],185, No. 12, 49–64 (1994).
V. N. Kolokol'tsov, “Geodesic flows on two-dimensional manifolds with additional first integral polynomial in the velocities,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],46, No. 5, 994–1010 (1982).
I. K. Babenko and N. N. Nekhoroshev, “Complex structures on two-dimensional tori admitting metrics with a nontrivial quadratic integral,”Mat. Zametki [Math. Notes],58, No. 5, 643–652 (1995).
M. L. Byalyi, “First integrals polynomial in the momenta for a mechanical system on the two-dimensional torus,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],21, No. 4, 64–65 (1987).
V. V. Kozlov and D. V. Treshchev, “Integrability of Hamiltonian systems with toroidal configuration space,”Mat. Sb. [Math. USSR-Sb.],135 (177), No. 1, 119–138 (1988).
V. V. Kozlov, “Periodic trajectories of a charge in a magnetic field,” in:Regular and Chaotic Dynamics (to appear).
A. Hurvits and R. Courant,Theory of Functions [Russian translation], Nauka, Moscow (1968).
V. V. Kozlov,Symmetries, Topology, and Resonances in Hamiltonian Mechanics [in Russian], Udmurt State Univ., Izhevsk (1995).
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Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 37–44, July, 1998.
The author wishes to express his thanks to V. V. Kozlov for his interest and his help in this work.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00747.
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Denisova, N.V. Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus. Math Notes 64, 31–37 (1998). https://doi.org/10.1007/BF02307193
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DOI: https://doi.org/10.1007/BF02307193