zbMATH — the first resource for mathematics

Topology of isoenergetic surfaces and bifurcation diagrams of integrable cases of dynamics of a rigid body on \(\text{SO}(4)\). (Russian) Zbl 0648.58016
The properties of the Hamiltonian systems defined on the Lie algebra of the \(\text{SO}(4)\) group are studied. These systems describe the dynamics of the four-dimensional rigid bodies with fixed centre of mass. The Hamiltonian \(H\) of a system of rigid bodies is written and the corresponding isoenergetic surface \(Q^ 3\) with Bott integral \(f\) is given by a graph \(\Gamma(Q,f)\). It is shown that the orbits \(S^ 2\times S^ 3\) in \(\text{SO}(4)\) are fibrations on the surface \(Q^ 3=\{H|_{S^ 2\times S^ 3= \text{const.}}\}\) and the integral \(f|_{S^ 2\times S^ 3}\) is of Bott type for every \(Q^ 3,\) except a finite number of such surfaces. The table of all possible graphs \(\Gamma(Q,f)\) associated to the Hamiltonian \(H\) is also given.

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures