I. Four equivalent properties of integrable billiards. (23/12)
Optical properties of conics have been known since the classical antiquity (and, according to the legend, put to use by Archimedes by destroying enemy ships with fire). The reflection in an ideal mirror is also known as the billiard reflection and, in modern terms, the billiard inside in ellipse is completely integrable. The interior of an ellipse is foliated by confocal ellipses that are its caustics: a ray of light tangent to a caustic remains tangent to it after reflection (“caustic” means burning).
I shall explain these classic results and some of their geometric consequences, including the Ivory lemma asserting that the diagonals of a curvilinear quadrilateral made by arcs of confocal ellipses and hyperbolas are equal. This lemma is in the heart of Ivory's calculation of the gravitational potential of a homogeneous ellipsoid.
I shall also describe the string construction that reconstructs a billiard table from its caustic; in particular, the string construction on an ellipse yields a confocal ellipse (a theorem of Graves). Then I will describe four equivalent properties of integrable billiards on Riemannian surfaces, connecting together the Ivory lemma, the string construction, billiard caustics, and Liouville metrics. The latter are generalizations of the metric of an ellipsoid in Euclidean space whose geodesic flows are completely integrable.
I shall conclude with a generalization of the famous Birkhoff's conjecture: If an interior neighborhood of a closed geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is a coordinate line.
II. Introducing symplectic billiards. (25/12)
I shall introduce a new dynamical system called symplectic billiards. As opposed to the usual (Birkhoff) billiards, where length is the generating function, for symplectic billiards the area is the generating function. I shall describe basic properties and exhibit several similarities, but also differences, of symplectic billiards with Birkhoff billiards. Symplectic billiards can be defined not only in the plane, but also in linear symplectic spaces, with the symplectic area as the generating function. In this multi-dimensional setting, I shall discuss the existence of periodic trajectories and describe the integrable dynamics of symplectic billiards in ellipsoids. I shall also explore polygonal symplectic billiards and present such polygonal billiards in which all orbits are periodic. This later work, still in progress, is mostly experimental, using specially developed computer programs.