New integrable cases in the dynamics of rigid bodies. II (Q1090474)

In a previous work [the author, ibid. 13, 169-172 (1986; Zbl 0606.70008)] we have introduced four new integrable cases of the problem of motion of a rigid body about a fixed point. The complete solution of the equations of motion in these cases reduce, in principle, to quadratures by virtue of Liouville's theorem. However, the direct procedure suggested by this theorem is rarely effective. In the majority of cases we need to seek for a suitable approach to the solution of each problem. In the present work we introduce a new integrable case which generalizes the last one of the previous work. The body is assumed to have complete dynamical symmetry at the fixed point. The forces admit no axial symmetry and can be interpreted as due to the combined interaction of mass, electric charges and magnetized parts of the body with external gravitational, electric and magnetic fields. The integration of the problem in this case is reduced to quadratures and a simple geometric interpretation of the motion is given. The simplest case is shown to be solvable in terms of the Jacobi Theta Functions.