The rigid body dynamics: classical and algebro-geometric integration

The basic notion for a motion of a heavy rigid body fixed at a point in three-dimensional space as well as its higher-dimensional generalizations are presented. On a basis of Lax representation, the algebro-geometric integration procedure for one of the classical cases of motion of three-dimensional rigid body - the Hess-Appel'rot system is given. The classical integration in Hess coordinates is presented also. For higher-dimensional generalizations, the special attention is paid in dimension four. The L-A pairs and the classical integration procedures for completely integrable four-dimensional rigid body so called the Lagrange bitop as well as for four-dimensional generalization of Hess-Appel'rot system are given. An $n$-dimensional generalization of the Hess-Appel'rot system is also presented and its Lax representation is given. Starting from another Lax representation for the Hess-Appel'rot system, a family of dynamical systems on $e(3)$ is constructed. For five cases from the family, the classical and algebro-geometric integration procedures are presented. The four-dimensional generalizations for the Kirchhoff and the Chaplygin cases of motion of rigid body in ideal fluid are defined. The results presented in the paper are part of results obtained in last decade.