Topology of isoenergetic surfaces and bifurcation diagrams of integrable cases of dynamics of a rigid body on \(\text{SO}(4)\). (Russian) Zbl 0648.58016

Oshemkov, A. A.

Usp. Mat. Nauk 42, No. 6(258), 199-200 (1987).

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.

Reviewer: Gheorghe Zet (Iaşi)

MSC:

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

Keywords:

isoenergetic surface; Bott integral; graph method; Hamiltonian systems; rigid bodies