Two Applications of Jacobi
نویسندگان
چکیده
We present new proofs of two results on the billiard ball problem by Rychlik R] and Bialy B]. x0. INTRODUCTION. We will give new proofs of two results on the billiard ball problem by Rychlik R] and Bialy B]. The original proofs were based on variational considerations. In our approach the variational context is absent, the dynamical system takes the center stage. We hope the simpliications provided by our method will make possible some progress on the conjectures for which these results lend partial support. x1.THE DYNAMICAL SYSTEM. Let us consider a convex domain Q in the plane. The billiard ball system is the ow t on Q S 1 deened by the free motion of a point particle in Q, with elastic reeections at the boundary @Q (the angle of reeection equal to the angle of incidence). The circle S 1 represents unit velocities. Strictly speaking, we need to identify the velocities at the boundary according to the collision law. The ow t preserves the Liouville measure equal to the product of the Lebesgue measures in Q and S 1. Birkhoo Bir] thought that this dynamical system is a very good model for Hamil-tonian dynamics. In the last thirty years his belief proved to be strikingly accurate. We understand as much about the low dimensional Hamiltonian dynamics, as we know about the billiard system. The ow t has a natural section map T : M ! M, where M = @Q 0; ]. The map T describes the dynamics \from collision to collision". The space M is the set of unit tangent vectors attached at the boundary and pointing inwards. It can be coordinatized by (s; '), where s is the arclength parameter taken modulo p, We would like to thank Gil Bor, Jian Cheng, Victor Donnay and Lenny Friedlander for helpful and enlightening discussions. The suggestions of the referee are also gratefully acknowledged.
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تاریخ انتشار 1993