# Convex billiards and non-holonomic systems. Fifty-First Annual DeLong Lectures. University of Colorado at Boulder . October 14, 2014

Invited Talk

Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For 'most' convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other curves, besides ellipses, for which this is true. In this talk, I'll discuss these phenomena and show how they are related to the geometry of nonholonomic plane fields (which will be defined and described).

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- October 14, 2014

### Service or Event Name

- Fifty-First Annual DeLong Lectures

### Host Organization

- University of Colorado at Boulder

### Location or Venue

- University of Colorado at Boulder