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Topological analysis of classical integrable systems in the dynamics of the rigid body. (English. Russian original) Zbl 0561.58021
Sov. Math., Dokl. 28, 802-805 (1983); translation from Dokl. Akad. Nauk SSSR 273, 1322-1325 (1983).
In this note the author describes the bifurcations that take place in the topological structure of the integral manifolds for the equations of motion of a rigid body around a fixed point under the assumption of complete integrability. The phase space is given by the manifold \(M=S^ 2\times {\mathbb{R}}^ 3\), and it is assumed that the system has three first integrals H,K,G: \(M\to {\mathbb{R}}\); two such cases are considered: the Kovalevskaya and the Goryachev-Chaplygin cases. The bifurcation set \(\Sigma\) of the mapping \(N=H\times K\times G: M\to {\mathbb{R}}^ 3\) is described, together with the topological modifications that the integral manifolds \(J_ n=\{z\in M|\) \(N(z)=n\}\) undergo as \(n\in {\mathbb{R}}^ 3\) crosses \(\Sigma\).
Reviewer: A.Vanderbauwhede

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
70E15 Free motion of a rigid body
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.)