On the integrability of the equations of dynamics in a non-potential force field

A range of issues related to the integration of the equations of motion of mechanical systems in non-potential force fields (often called circulatory systems) are discussed. The approach to integration is based on the Euler–Jacobi–Lie theorem: for exact integration of a system with $n$ degrees of freedom it is necessary to have $2n-2$ additional first integrals and symmetry fields (taking the conservation of the phase volume into account) which are in certain natural relations to one another. The cases of motion in non-potential force fields that are integrable by separation of variables are specified. Geometric properties of systems with non-Noether symmetry fields are discussed. Examples of the existence of irreducible polynomial integrals of the third degree in the momentum are given. The problem of conditions for the existence of single-valued polynomial integrals of circulatory systems with two degrees of freedom and toric configuration spaces is considered. It is shown that in a typical case the equations of motion do not admit non-constant polynomial integrals.
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