Today I showed some students how Lie Brackets have the interesting ability to reveal hidden trajectories which can be assembled through motion primitives, resulting in non-holonomic approximations of holonomic movement. Parallel parking is, of course, the most famous case of this.
A student asked if there were other examples of Lie Brackets revealing surprising new control vectors. I thought that was a very good question, but I was at a loss to give any good examples beyond parallel parking and a rolling ball.
Any ideas?