On the Integrability of Circulatory Systems

    2022, Volume 27, Number 1, pp.  11-17

    Author(s): Kozlov V. V.

    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.
    Keywords: circulatory system, polynomial integral, genus of surface
    Citation: Kozlov V. V., On the Integrability of Circulatory Systems, Regular and Chaotic Dynamics, 2022, Volume 27, Number 1, pp. 11-17

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