Geometry and Billiards (Student Mathematical Library)

By Serge Tabachnikov

Mathematical billiards describe the movement of a mass element in a website with elastic reflections off the boundary or, equivalently, the habit of rays of sunshine in a site with preferably reflecting boundary. From the perspective of differential geometry, the billiard circulation is the geodesic move on a manifold with boundary. This ebook is dedicated to billiards of their relation with differential geometry, classical mechanics, and geometrical optics. themes lined comprise variational ideas of billiard movement, symplectic geometry of rays of sunshine and imperative geometry, lifestyles and nonexistence of caustics, optical homes of conics and quadrics and entirely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known topic of twin (or outer) billiards. The ebook is predicated on a sophisticated undergraduate themes direction. minimal necessities are the normal fabric coated within the first years of faculty arithmetic (the whole calculus series, linear algebra). besides the fact that, readers should still convey a few mathematical adulthood and depend upon their mathematical logic. a distinct characteristic of the ebook is the insurance of many assorted issues on the topic of billiards, for instance, evolutes and involutes of airplane curves, the four-vertex theorem, a mathematical conception of rainbows, distribution of first digits in quite a few sequences, Morse conception, the Poincaré recurrence theorem, Hilbert's fourth challenge, Poncelet porism, and so on. There are nearly a hundred illustrations. The ebook is appropriate for complicated undergraduates, graduate scholars, and researchers drawn to ergodic concept and geometry. This quantity has been copublished with the math complex examine Semesters software at Penn country.

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Manuscripta Mat. 108 (2002), 431–437. [37] M. Farber, S. Tabachnikov. Topology of cyclic configuration areas and periodic orbits of multi-dimensional billiards. Topology forty-one (2002), 553– 589. [38] R. Fox, R. Kershner. Geodesics on a rational polyhedron. Duke Math. J. 2 (1936), 147–150. [39] G. Galperin. Billiard balls count number π, MASS Selecta, Amer. Math. Soc. , windfall, RI, 2003, pp. 197–204. [40] G. Galperin. Convex polyhedra with no easy closed geodesics. Reg. Chaotic Dynamics eight (2003), 45–58. [41] G. Galperin, N. Chernov.

15 implies the Poncelet porism. As was once defined in bankruptcy four, the closed invariant curves of an integrable quarter conserving transformation hold an affine constitution, during which the transformation is a translation x → x+c the place c relies on the curve. particularly, the map is periodic on a curve if and provided that c ∈ Q (independently of the purpose x). We finish this bankruptcy with a dialogue of multi-dimensional twin billiard; see [105, 108, 107, 113]. One desires to change the twin billiard curve via a tender strictly convex closed hypersurface M in a vector house and use tangent traces to M to outline a twin billiard map.

Attach the 2 maxima through a curve inside of M and view the minimal of L in this curve. Take the utmost of those minima over all such curves. this can be additionally a severe aspect of L, except the maxima; see determine 6. four. A sophisticated aspect is to teach that this severe element lies now not at the boundary of M . This follows from the very fact, illustrated in determine 6. three, that the functionality L raises as one strikes from the boundary. determine 6. four. Mountain cross kind severe element The argument is illustrated, for n = 2, via determine 6.

6) converges, okay (ak + bk ) < C for a few consistent C. this means that the second one summand in (5. 7) is arbitrarily small for sufficiently huge m. It follows that, for giant m, the functionality fm has as many signal alterations because the n-th harmonic, that's, 2n, and we're performed. 2) allow us to argue through contradiction. think that f has lower than 2n signal alterations at the circle. The variety of signal adjustments being even, f has at such a lot 2(n − 1) of them. you will see that a trigonometric polynomial g of measure ≤ n − 1, that's, n−1 g(φ) = (ak cos okayφ + bk sin kφ) , k=0 that alterations symptoms accurately within the similar issues as f .

The gap of traces within the affine aircraft is received from the distance of traces within the projective airplane through deleting the road at infinity. therefore the previous house is (RP2 )∗ with some extent deleted that is, topologically, an open Moebius band; see workout five. 21. Projective duality extends to delicate curves within the related means as mentioned above for Euclidean aircraft; specifically, the correspondence 90 five. life and Non-existence of Caustics among a variety of singularities, depicted in determine five. 12, nonetheless holds.

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