The Serret-Andoyer formalism in rigid-body dynamics: I. Symmetries and perturbations

This paper reviews the Secret-Andoyer (SA) canonical formalism in rigid-body dynamics, and presents some news results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, that stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are not longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Secret-Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specially, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antimony between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into action-angle ones, using the method of Sadov.