Billiard Algebra, Integrable Line Congruences, and Double Reflection Nets

Abstract

Billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Sixpointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrability condition of a line congruence. A new notion of the double reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properties and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are defined and discussed.

Heisenberg Model in Pseudo-Euclidean Spaces II

“…Thus, the segments x k−1 x k and x k x k+1 determined by 3 successive points of the mapping (2.3), (2.4) may be either on the same side of the tangent plane T x k Q n−1 (the usual billiard reflection at x k ), or on the opposite sides of T x k Q n−1 . Such configurations were studied in [3,4,6,12].…”

“…Since {φ 1 , φ 2 } = 4 A −1 x, y = 0| M c,h , it follows that M c,h is a symplectic submanifold of R 2n (x, y) and the mapping φ is a symplectic transformation of M c,h (see Theorem 2.1, [18] 5 ). 6 The Hamiltonian and contact integrability of the virtual billiard dynamics is described in [18]. In the case when EA is positive definite, c = +1, this is a billiard system within ellipsoid Q n−1 in the pseudo-Euclidean space (see [19,5]).…”

“…Recall that if some of the eigenvalues of the matrix given for c = 1, but the same proof applies for c = 0 and c = −1. 6 Here, as in the third footnote we note that for the codomain of φ we should consider the variety M c,h without the assumptions A −2 x, x = 0, A −1 y, y = 0.…”

Pseudo-integrable billiards and double reflection nets

“…Теорема о двойном отражении является прективно-геометрическим стержнем динамики биллиардов в областях, ограниченных софокусными квадриками; она утверждает, что отражения от двух софокусных квадрик можно переставлять. Четыре прямые, получающиеся отражением друг друга в контексте теоремы о двойном отражении, образуют конфигурацию двойного отражения, играющую роль квад-уравнения для недавно открытого класса интегрируемых конгруэнций прямых, называемых решетками двойных отражений [20]. Интегрируемость возникает как следствие операционной непротиворечивости биллиардной алгебры из [16]; см.…”

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

“…In Euclidean spaces of arbitrary dimension, such configurations were introduced by Dragović and Radnović in [9]. It appears that a multidimensional variant of Darboux's 4-periodic virtual trajectory with reflections on two quadrics, refereed as double-reflection configuration [11], is fundamental in the construction of the double reflection nets in Euclidean spaces (see [13]) and in pseudo-Euclidean spaces (see [14]). They also played a role in a construction of the billiard algebra in [10].…”