Geometry of Integrable Billiards and Pencils of Quadrics

We study the deep interplay between geometry of quadrics in d-dimensional space and the dynamics of related integrable billiard systems. Various generalizations of Poncelet theorem are reviewed. The corresponding analytic conditions of Cayley's type are derived giving the full description of periodical billiard trajectories; among other cases, we consider billiards in arbitrary dimension d with the boundary consisting of arbitrary number k of confocal quadrics. Several important examples are presented in full details demonstrating the effectiveness of the obtained results. We give a thorough analysis of classical ideas and results of Darboux and methodology of Lebesgue, and prove their natural generalizations, obtaining new interesting properties of pencils of quadrics. At the same time, we show essential connections between these classical ideas and the modern algebro-geometric approach in the integrable systems theory.Comment: 49 pages, 14 figure

Explicit solutions to Poncelet’s porism

AbstractSolution to the following problem is considered: for given conics C and K and an integer N⩾3, determine whether there exists a closed N-sided polygon inscribed in C and circumscribed about K. The case of C and K being circles is considered in detail. Equations are proposed with a relatively small number of arithmetic operations – near log2N. Along the way, the following result is obtained: for circles with rational coefficients, the polygons can only have the following number of sides N=3, 4, 5, 6, 8, 10 and 12 (a subset of the Mazur’s set of integers for rational elliptic curves). The proposed solution may also be applied to determine whether a Hankel determinant of order N/2 having special form (used in the classical Cayley criterion) is equal to zero, and for related problems. Possible generalizations for ellipses and hyperbolas are also presented. Particularly, equations are proposed for parameters of concentric Poncelet’s ellipses (billiard case)

How Discrete Patterns Emerge from Algorithmic Fine-Tuning: A Visual Plea for Kroneckerian Finitism

International audienceThis paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker's conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker's main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical understanding prevail over mere preemptive reductionism to whole numbers

Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.Comment: 46 pages, 4 figures; minor revisions from v1; accepted for publication in Adv. Mat

Nonpersistence of resonant caustics in perturbed elliptic billiards

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.Comment: 14 pages, 3 figure