[Paper Review] On the Motion of Billiards in Ellipses

Hellmuth Stachel|arXiv (Cornell University)|May 8, 2021

Mathematics and Applications5 references4 citations

TL;DR

This paper establishes a canonical parametrization of billiard motions in ellipses using Jacobian elliptic functions with the numerical eccentricity of the caustic as modulus. By analyzing the kinematics of billiard motion, it shows that the billiard transformation acts as a coordinate shift, enabling explicit parametrization of the billiard and associated Poncelet grids, and derives invariants involving side lengths and distances to the caustic, such as constant products of distances over periodic orbits.

ABSTRACT

For billiards in an ellipse with an ellipse as caustic, there exist canonical coordinates such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A kinematic analysis of billiard motions paves the way to an explicit canonical parametrization of the billiard and even of the associated Poncelet grid. This parametrization uses Jacobian elliptic functions to the numerical eccentricity of the caustic as modulus.

Motivation & Objective

• To develop a canonical coordinate system for billiard motions in ellipses with confocal elliptical caustics.
• To establish a connection between billiard dynamics and Jacobian elliptic functions using the caustic's numerical eccentricity as the modulus.
• To provide an explicit parametrization of the billiard trajectory and the associated Poncelet grid using elliptic functions.
• To derive and prove invariants in periodic billiard motions, particularly concerning distances from vertices to the caustic and side lengths.

Proposed method

• Utilizes a kinematic analysis of billiard motion to derive an infinitesimal transformation preserving confocal ellipses and permuting confocal hyperbolas and caustic tangents.
• Introduces a canonical parameter $ u $ via integration of the infinitesimal transformation, leading to a group of transformations with elliptic function structure.
• Employs Jacobian elliptic functions $ ext{dn} $ with modulus $ m $, the numerical eccentricity of the caustic, to parametrize the billiard motion.
• Derives the parametrization of the ellipse $ e $ and the associated Poncelet grid using $ ext{dn} $ functions and the parameter $ ilde{u} $, linked to the elliptic integral of the first kind.
• Applies the parametrization to analyze the variation of side lengths and distances to the caustic under billiard motion.
• Uses differentiation and symmetry arguments to prove invariance of products of distances and side lengths over periodic orbits.

Experimental results

Research questions

• RQ1 How can billiard motions in ellipses with elliptical caustics be parametrized using elliptic functions?
• RQ2 What is the role of the numerical eccentricity of the caustic as the modulus in constructing canonical coordinates for the billiard motion?
• RQ3 How do the distances from billiard vertices to the caustic and side lengths behave under periodic motion?
• RQ4 What invariants emerge from the parametrization, particularly for $ N $-periodic billiards with even $ N $?
• RQ5 Can the Poncelet grid associated with a periodic billiard be explicitly mapped using elliptic functions?

Key findings

• The billiard transformation from vertex to vertex is equivalent to a shift in the canonical parameter $ u $, which is parametrized using Jacobian elliptic functions with modulus equal to the numerical eccentricity of the caustic.
• The parametrization of the ellipse $ e $ and the associated Poncelet grid is explicitly given in terms of $ ext{dn} $ functions, enabling a complete description of the motion.
• For $ N $-periodic billiards with even $ N $, the products $ r_i r_{i+n} = l_i l_{i+n} = k_e $ hold, where $ r_i $ and $ l_i $ are distances from vertex $ P_i $ to the contact points on the caustic.
• The product of all $ r_i $ or $ l_i $ over a full period is invariant and equals $ k_e^{N/2} $, with $ k_e^{N/4} $ for $ N \equiv 0 \pmod{4} $.
• The ratio of side lengths satisfies $ s_{i+n}/s_i = l_{i+n}/r_{i+1} = r_{i+n+1}/l_i $ for $ N=4n $, and similar relations for $ N=4n+2 $, confirming structural invariance.
• The sum $ \sum \cos \theta_i $ is invariant under billiard motion, confirming a known result with a new derivation via parametrization and differentiation.