Discrete symmetries of dynamics equations with polynomial integrals of higher degrees

In the article, systems with toric configuration space and kinetic energy in the form of a “flat” Riemannian metric on the torus are considered.The potential energy $V$ is a smooth function on the configuration torus.The dynamics of such systems are described by “natural” Hamiltonian systems of differential equations. If we replace $V$ with $\epsilon V$, where $\epsilon$ is a small parameter, then, according to Poincare, the study of such Hamiltonian systems at small values of $\epsilon$ refers to the “main problem of dynamics”. The paper discusses a well-known hypothesis about unambiguous momentum-polynomial integrals of the equations of motion: if there is a momentum-polynomial integral of degree $m$, then there will necessarily be a linear or quadratic momentum first integral. It is fully proved for $m=3$ and $m=4$. The cases of “higher” degrees when $m=5$ and $m=6$ are discussed as well.Following the theory of Hamiltonian systems’ perturbations, we introduce resonant lines on the plane of impulses. If the system allows for a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonant lines are found, which gives, in particular, the necessary conditions for integrability. Some new criteria for the existence of unambiguous polynomial integrals are obtained.

configuration torus, Hamiltonian system, spectrum, resonant lines, polynomial integrals