Abstract:
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean
plane is presented. The novelty of the approach is based on a relationship recently established
by the authors between periodic billiard trajectories and extremal polynomials on the systems
of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space.
Even in the planar case systematically studied in the present paper, it leads to new results
in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories,
which are $n$-periodic in elliptical coordinates. The characterizations are done both in terms
of the underlying elliptic curve and divisors on it and in terms of polynomial functional
equations, like Pell's equation. This new approach also sheds light on some classical results.
In particular, we connect the search for caustics which generate periodic trajectories with
three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer.
The main classifying tool are winding numbers, for which we provide several interpretations, including
one in terms of numbers of points of alternance of extremal polynomials. The latter implies
important inequality between the winding numbers, which, as a consequence, provides another
proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with
small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics.
As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly
separable polynomials has been observed for all those small periods.
References