On the integrability of Hamiltonian systems with toral position space

This paper considers the problem on the complete integrability of a Hamiltonian system with a toral position space, with Euclidean kinetic energy and a small analytic potential. Necessary integrability conditions are found in the case when the potential is a trigonometric polynomial. These conditions are also necessary conditions of existence of additional first integrals, polynomial in the momenta (with no assumption on the smallness of the potential). The proofs are based on a detailed analysis of the classical scheme of perturbation theory. The general results are applied to the study of the complete integrability of the well-known problem on the motion of $n$ points along a line with periodic interaction potential. In particular, the nonintegrability of the “open” chain of interactions of particles is proved for $n>2$; the “periodic” chain is nonintegrable with the additional condition that the potential be a nonconstant trigonometric polynomial. Conditions for complete integrability of the generalized nonperiodic Toda chain are discussed.
Bibliography: 17 titles.