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On the integrability of Hamiltonian systems with toral position space. (English. Russian original) Zbl 0696.58022
Math. USSR, Sb. 63, No. 1, 121-139 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 1, 119-138 (1988).
Summary: This paper considers the problem of 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 of 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.

##### MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H05 Hamilton’s equations
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