The Falling Cat problem is a lovely example of the application of gauge theory to problems in mechanics and control. Richard Montgomery's beautiful "Falling Cat Theorem," relates optimal reorientation of the falling cat to the dynamics of particles in Yang-Mills fields. On his webpage, he states,

"The cat flips itself right side up, even though its angular momentum is zero. It does this by changing its shape. In terms of gauge theory, the shape space of the cat forms the base space of a principal SO(3)-bundle, and the statement "angular momentum equals zero" defines a connection on this bundle."

If you know where I might find a print of Gerard Lacz's awesome photo of a falling cat, which is pictured on the left, please let me know.