Geometry and Billiards (Student Mathematical Library)
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Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differential geometry, classical mechanics, and geometrical optics. Topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual (or outer) billiards. The book is based on an advanced undergraduate topics course. Minimum prerequisites are the standard material covered in the first two years of college mathematics (the entire calculus sequence, linear algebra). However, readers should show some mathematical maturity and rely on their mathematical common sense. A unique feature of the book is the coverage of many diverse topics related to billiards, for example, evolutes and involutes of plane curves, the four-vertex theorem, a mathematical theory of rainbows, distribution of first digits in various sequences, Morse theory, the Poincaré recurrence theorem, Hilbert's fourth problem, Poncelet porism, and many others. There are approximately 100 illustrations. The book is suitable for advanced undergraduates, graduate students, and researchers interested in ergodic theory and geometry. This volume has been copublished with the Mathematics Advanced Study Semesters program at Penn State.
1-parameter family of parallel periodic trajectories of the same period and length, and an odd-periodic one is contained in a strip consisting of trajectories whose period and length is twice as great; see figure 7.2. 113 114 7. Billiards in Polygons B R P O A Q C Figure 7.1. Fagnano billiard trajectory in an acute triangle Figure 7.2. A strip of parallel periodic billiard trajectories Exercise 7.2. a) Let P be a convex quadrilateral that has a 4periodic “Fagnano” billiard trajectory
faces of M . One has the Euler formula: v − e + f = 2. Let us compute the sum S of all angles of the faces of M . At a vertex, the sum of angles is 2π − k where k is the curvature of this vertex. Summing up over the vertices gives: (7.1) S = 2πv − K where K is the total curvature. On the other hand, one may sum over the faces. The sum of the angles of the i-th face is π(ni − 2), where ni is the number of sides of this face. Hence (7.2) S=π ni − 2πf. Since every edge is adjacent to two faces,
the billiard inside a cone whose faces are convex inside and satisfy certain geometrical conditions (cf. figure 1.4 and model Example 1.10 in Chapter 1). An analog of Theorem 7.17 holds for such systems as well. This result was recently obtained by D. Burago, S. Ferleger and A. Kononenko using ideas of Alexandrov’s geometry; see, e.g.,  for a survey. Let us formulate one of their theorems: the number of collisions of n elastic balls in space with masses m1 ≥ · · · ≥ mn does not exceed 400n 2
properties usually associated with chaos: sensitivity to initial conditions, density of periodic orbits, density of the orbit of any open set, etc.1 The first examples of billiards with hyperbolic dynamics were discovered by Ya. Sinai : these billiards are bounded by piecewise smooth curves whose smooth components are strictly convex inwards and which intersect transversally. See figure 1.5, a torus or a square with a convex hole, and figure 8.2. A parallel beam of light, after a reflection
anisotropic). Consider the level curves of the function v and let γ be a trajectory of light in this medium. Let t be the speed of light along γ considered as a function on this curve. Denote by 2Incidentally, the cycloid also solves another problem: to find a curve AB such that a mass point, sliding down the curve, arrives at the end point B in the same time, no matter where on the curve it started. 18 1. Motivation: Mechanics and Optics A B Figure 1.13. Brachistochrone α(t) the angle