On the Integrability of Circulatory Systems
2022, Volume 27, Number 1, pp. 11-17
Author(s): Kozlov V. V.
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.
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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