# 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.

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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