# 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