Quasi-periodic Motions of a Rigid Body. I. Quadratic Hamiltonians on the Sphere with a Distinguished Parameter

The motion of a dynamically symmetric rigid body, fixed at one point and subject to an affine (constant+linear) force field is studied. The force being weak, the system is treated as a perturbation of the Euler top, a superintegrable system. Averaging along the invariant $2$-tori of the Euler top yields a normal form which can be reduced to one degree of freedom, parametrized by the corresponding actions. The behaviour of this family is used to identify quasi-periodic motions of the rigid body with two or three independent frequencies.