On the motion of billiards in ellipses

Abstract

For billiards in an ellipse e with an ellipse as caustic, there exist canonical coordinates on e such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A kinematic analysis of billiard motions offers a new approach to canonical parametrizations of billiards and associated Poncelet grids. This parametrization uses Jacobian elliptic functions with the modulus equal to the numerical eccentricity of the caustic and is the basis for proving a few invariants of periodic billiards.

Exploring self-intersected N-periodics in the elliptic billiard

“…Points of self-intersections of Poncelet 𝑁 -periodics are located on confocal conics of the associated Poncelet grid [16,22,25]. A kinematic analysis of the geometry of 𝑁 =periodics using Jacobian elliptic functions is proposed in [24]. Works [13,19] derive explicit expressions for some invariants in the 𝑁 = 3 case (billiard triangles).…”

“…Here we focus on trajectories which are self-intersected, i.e., which wrap around the inner conic, or caustic, more than once (i.e., their turning number is greater than one [24]). Figure 1 (resp.…”

Mather β-Function for Ellipses and Rigidity

“…There is an extensive literature on Poncelet porism, formulas for invariant measure and the rotation number. I refer to the incomplete list of papers on the subject [ 1 , 9 , 10 , 11 , 12 , 13 ].…”

“…The non-standard generating function for convex billiards has been already used in our paper [ 14 ], explaining conservation laws for elliptical billiards discovered recently by Dan Reznik [ 15 , 16 ] et al, see also [ 11 , 12 , 17 ]. Additionally, the non-standard generating function is a key ingredient in the recent proof of a part of Birkhoff conjecture for centrally symmetric billiard tables [ 18 ].…”

A Triple of Projective Billiards

“…A billiard is the trajectory of a mass point in a domain called billiard table with ideal physical reflections in the boundary. Already for two centuries, billiards in ellipses (see Figures 1,2,8) and their projectively equivalent counterparts have attracted the attention of mathematicians, beginning with J.-V. Poncelet [4] and C.G.J. Jacobi [3] and continued, e.g., by S. Tabachnikov, who addresses in his book [10] a wide variety of themes around this topic.…”