# 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.

##### 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