Parallel parking, falling cats, spherical robots and differential geometry by Prof. Ravi N Banavar

The notion of "parallel parking a car" is known to many of us. Through a sequence of actions consisting of - steering (turning the steering wheel clockwise or anti-clockwise) and driving (moving the car forward or backward) - we successfully park the car in the available slot, though, at first sight, no lateral motion of the vehicle seemed possible with the permissible control. This skill, often picked up through intuition and experience, has a deeper mathematical interpretation as well - termed the the Lie bracket of vector fields. The Lie bracket builds on the familiar constructs that we learnt in high school about the cross product in the Euclidean space R^3, and commutativity of matrices. In this talk, apart from the example just stated, I shall introduce a few others - locomotion of spherical robots, the tumbling of a spacecraft or the vertical fall of a cat - all of which have a common underlying theme, namely, that their evolution (dynamics) is explained with differential geometric constructs.In the process, I shall briefly introduce the fascinating subject of differential geometry, from an engineer's perspective.