Billiards in ellipses revisited
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Billiards in ellipses revisited Arseniy Akopyan1 · Richard Schwartz2 · Serge Tabachnikov3
Received: 27 January 2020 / Revised: 27 January 2020 / Accepted: 10 August 2020 © Springer Nature Switzerland AG 2020
Abstract We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods. Keywords Elliptic billiards · Complete integrability · Poncelet polygon · Poncelet grida · Minkowski billiards Mathematics Subject Classification 37J35 · 37C83 · 51M15
1 Introduction The billiard in an ellipse is a thoroughly studied completely integrable dynamical system, see, e.g., [18]. In particular, a billiard trajectory that is tangent to a confocal ellipse will remain tangent to it after each reflection. That is, the confocal ellipses are the caustics of this billiard.
AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). RS is supported by NSF Grant DMS-1807320. ST was supported by NSF Grants DMS-1510055, DMS-2005444, and SFB/TRR 191.
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Serge Tabachnikov [email protected] Arseniy Akopyan [email protected] Richard Schwartz [email protected]
1
Institute of Science and Technology, 3400 Klosterneuburg, Austria
2
Department of Mathematics, Brown University, Providence, RI 02912, USA
3
Department of Mathematics, Penn State University, University Park, PA 16802, USA
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A. Akopyan et al.
Fig. 1 Angles of a billiard trajectory
One of the properties of this system, a consequence of its complete integrability, is that a periodic billiard trajectory tangent to a confocal ellipse includes into a 1parameter family of periodic trajectories tangent to the same confocal ellipse and having the same period and the same rotation number. This is the assertion of the Poncelet porism for confocal ellipses. The Poncelet porism concerns the same kind of 1-parameter family of polygons that are simultaneously inscribed and circumscribed in the same pair of conics, but in general the conics need not be confocal. A classic result about a continuous 1-parameter family of billiard paths is that their perimeters remain constant. (See [18], and also Lemma 2.3 below.) Recently Dan Reznik conducted a large series of computer experiments with periodic orbits in elliptic billiards and discovered numerous new properties of these polygons that are similar in spirit to the constant-perimeter result. See [6,9–15]. In this paper we give proofs which verify some of these observations. Essentially, we prove three main results. In the body of the paper, we will also prove a number of variants and generalizations. We would also like to mention that another proof of the first two results is presented in [4]; it is based on a non-standard generating function for convex b
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