Joint Colloquium: Bryant -- Convex Billiards and Nonholonomic Systems

Speaker: Prof Robert Bryant

http://fds.duke.edu/db/aas/math/bryant

Time: Thursday, October 10, 2:00--3:00PM

Room: Chemistry Lecture Theatre 4, University of Sydney


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Title: Convex Billiards and Nonholonomic Systems

Abstract:

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 convex curves, besides ellipses, for
which there are infinitely many closed billiard n-gons for some n.  In this talk, I’ll
discuss the above-mentioned phenomenon and show how it is related to the geometry of
non-holonomic plane fields (which will be defined and described).  This leads to some
surprisingly beautiful geometry, which will require nothing beyond multivariable
calculus from the audience.


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Joint Colloquium web site: 

http://www.maths.usyd.edu.au/u/SemConf/JointColloquium/index.html