×

zbMATH — the first resource for mathematics

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.

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