Abstract
This paper discusses conditions for the existence of polynomial (in velocities) first integrals of the equations of motion of mechanical systems in a nonpotential force field (circulatory systems). These integrals are assumed to be single-valued smooth functions on the phase space of the system (on the space of the tangent bundle of a smooth configuration manifold). It is shown that, if the genus of the closed configuration manifold of such a system with two degrees of freedom is greater than unity, then the equations of motion admit no nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with configuration space in the form of a sphere and a torus which have nontrivial polynomial laws of conservation. Some unsolved problems involved in these phenomena are discussed.
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This work was supported by a grant of RSF (project No. 21-71-30011).
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To the memory of Aleksey Borisov
MSC2010
37N05
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Kozlov, V.V. On the Integrability of Circulatory Systems. Regul. Chaot. Dyn. 27, 11–17 (2022). https://doi.org/10.1134/S1560354722010038
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DOI: https://doi.org/10.1134/S1560354722010038