The paper Geometry of the Prytz Planimeter described a mechanical instrument whose configuration space is an $S^1$-bundle with an $SU(1,1)$ action. That paper goes on to study the holonomies of various paths in the base space. Motion is described by a connection on this bundle.
I am interested because it's an example of a $G$-bundle appearing in classical mechanics. Are there other explicit classical mechanical systems that engineers study that exhibit the basic concepts of differential geometry so lucidly?