Classical and quantum mechanics of jointed rigid bodies with vanishing total angular momentum

A gauge theoretical treatment proves to have been successful in the study of systems of point particles; the center-of-mass system is made into a principal fiber bundle, on which is defined a natural connection. The gauge theoretical approach may be generalized to be applicable to a system of rigid bodies. The present article deals with a system of two identical axially symmetric cylinders jointed together by a special type of joint. This system is the model made by Kane and Scher and reformulated later by Montgomery, in order to study the falling cats who can land on their legs when released upside down. With the no-twist condition, the system turns out to have the configuration space diffeomorphic with SO(3), which is made into a principal O(2) bundle over RP2, the real projective space of dimension two, and endowed with a natural connection. An optimal control problem for this system with the vanishing total angular momentum is satisfactorily treated in this bundle picture. Along with a certain perform...