Differential geometry of the falling cat.

Everyone knows that a cat dropped upside downcan turn around and fall on his legs. This ability, which at first glance would seem to contradict the conservation of angular momentum, it is instead a consequence of it and is based on the cat's ability to change shape over the course of the fall. In the first part of the seminar we will discuss the kinematics of a deformable body (the cat, but it could also be a satellite or a robotic arm) from the points of view of differential geometryfollowing R. Montgomery. We will show that the configuration space of a deformable body has the structure of a principal bundle with structure group SO(3) -- the group of rotations of the three-dimensional Euclidean space -- and that the angular momentum defines a connection on this bundle. Finding paths with zero angular momentum thus becomes a problem of parallel transport. In the second part of the seminar we will apply this general theory to a model of cat introduced by Kane and Scher, always followingMontgomery.