Another couple of examples are the sphere rolling on the plane (or any surface, for that matter) without twisting or slipping, which is described by a connection on a principal SO(3)-bundle over the surface, and the motion of a trailer (or series of hitched trailers) being pulled by a tractor, which is described by a PSL(2,R)-connection (or a product of these) over the space of positions of the tractor. (This latter example is probably closely related to the planimeter problem described in Foote's article.)
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