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Statistical dynamics of a system of coupled pendulums. (English. Russian original) Zbl 1041.37504
Dokl. Math. 62, No. 1, 129-131 (2000); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 373, No. 5, 597-600 (2000).
The author considers a Hamiltonian system with Hamiltonian \(H=T+V\) with the \(n\)-torus as configuration space, where \(T\) is the kinetic energy and the potential energy \(V\) is assumed to be a trigonometric polynomial. Let \(\Sigma\) denote the convex hull of the spectrum of \(V\). An edge joining two vertices of the polyhedron \(\Sigma\) is said to be positive if \(\langle\alpha,\beta\rangle>0\) (scalar product in \( R^n\)). The main result is the following theorem: Let \(W\) be the hyperplane containing the origin and \(n-1\) linearly independent vertices of \(\Sigma\); if a positive edge not lying entirely in \(W\) adjoins each of these \(n-1\) vertices, then the Hamiltonian system does not admit a polynomial integral that is independent of the energy \(H\). This is proved using the results of the author and D. V. Treshchev [Mat. Sb., Nov.Ser. 135(177), No. 1, 119–138 (1988; Zbl 0696.58022)]. This result is applied to the following theorem: A system of \(n\) pendulums coupled by elastic springs does not admit a single-valued integral that is independent of the total energy. In particular, this enables the author to derive the Gibbs distribution of the system in an explicit form.

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H05 Hamilton’s equations