Orbits In Billiards Research Articles

Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries

We study the geometry of reflection of a massive point-like particle at conic section boundaries. Thereby the particle is subjected to a central force associated with either a Kepler or Hooke potential. The conic section is assumed to have a focus at the Kepler center, or have its center at the Hookian center respectively. When the particle hits the boundary it is ideally reflected according to the law of reflection. These systems are known to be integrable.We describe the consecutive billiard orbits in terms of their foci. We show that the second foci of these orbits always lie on a circle in the Kepler case. In the Hooke case, we show that the foci of the orbits lie on a Cassini oval. For both systems we analyze the envelope of the directrices of the orbits as well.

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

Birkhoff normal form and twist coefficients of periodic orbits of billiards* *X J is supported in part by the NSF Grant DMS-1854232.

In this paper we study the Birkhoff normal form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist coefficients in terms of the geometric parameters of the billiard table. As an application, we obtain characterizations of the nonlinear stability and local analytic integrability of the billiards around the elliptic periodic points.

Quantitative behavior of non-integrable systems. III

The main purpose of part (III) is to give explicit geodesics and billiard orbits in polysquares that exhibit time-quantitative density. In many instances, we can even establish a best possible form of time-quantitative density called superdensity. We also study infinite flat dynamical systems, both periodic and aperiodic, which include billiards in infinite polysquare regions. In particular, we can prove time-quantitative density even for aperiodic systems. In terms of optics the billiard case is equivalent to the result that an explicit single ray of light can essentially illuminate a whole infinite polysquare region with reflecting boundary acting as mirrors. In fact, we show that the same initial direction can work for an uncountable family of such infinite systems.

Unexpected linearly stable orbits in 3-dimensional billiards

In this work, we construct linearly stable periodic orbits in 3-dimensional domains with boundaries containing focusing components (small pieces of a sphere) where we place these components arbitrarily far apart. It demonstrates that we cannot directly implement the construction methods of planar billiards with focusing boundaries and chaotic properties into 3-dimensional billiards.

New Properties of Triangular Orbits in Elliptic Billiards

New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in “Can the elliptic billiard still surprise us?” (2020) Math. Intelligencer 42(1): 6–17, some of which were generalized to N > 3. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.

Billiards in ellipses revisited

We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.

Complex Caustics of the Elliptic Billiard

The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. In the case of 4-periodic orbits, we get generically three caustics. We also give an upper bound on the number of caustics for orbits with a fixed number of sides, and explain how to compute its exact value.

Stability of periodic orbits in no-slip billiards

Rigid bodies collision maps in dimension-two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards—planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change after collisions with the boundary of the billiard domain.This paper, which continues the investigation initiated in Cox and Feres (2017 Dynamical Systems, Ergodic Theory, and Probability: in Memory of Chernov (Providence, RI: American Mathematical Society), is mainly focused on the issue of stability of periodic orbits in no-slip planar billiards. We prove Lyapunov stability of periodic orbits in polygonal billiards of this kind and, for general billiards domains, we obtain curvature thresholds for linear stability at commonly occurring period-2 orbits. More specifically, we prove that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than π, no-slip billiard maps admit elliptic period-2 orbits; (ii) polygonal no-slip billiards under this same corner angle condition always contain small invariant neighborhoods of the periodic point on which, up to smooth conjugacy, orbits of the return map lie on concentric circles; in particular the system cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of the Sinai billiard must contain linearly stable periodic orbits of period 2 and, more generally, we obtain a curvature threshold at which the period-2 orbits go from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in wedge and triangular billiards. Our linear stability results extend those of Wojtkowski for the no-slip Sinai billiard.

λ-stability of periodic billiard orbits

We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is λ-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as . This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be λ-stable and prove that the set of d-gons having at most finite number of λ-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine completely the λ-stable periodic orbits in integrable polygons.

Scarcity of Periodic Orbits in Outer Billiards

We give a simple proof of the result of Tumanov and Zharnitsky (Int Math Res Not, 2006) that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a parallelogram. We also obtain results on period 5 and 6 orbits.

Linear and nonlinear stability of periodic orbits in annular billiards.

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary and also a circular scatterer in the interior of the disk. We investigate the stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of an arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of Kolmogorov-Arnold-Moser islands. Thus, we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.

Periodic orbits of oval billiards on surfaces of constant curvature

ABSTRACTIn this paper, we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a two-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary of the region is an oval. Using these properties and defining good perturbations for billiards, we show that having only a finite number of nondegenerate periodic orbits for each fixed period is an open property for billiards on surfaces of constant curvature and a dense one on the hyperbolic plane. We finish this paper studying the stability of these nondegenerate orbits.

Shortest closed billiard orbits on convex tables

Given a planar compact convex billiard table T, we give an algorithm to find the shortest generalised closed billiard orbits on T. (Generalised billiard orbits are usual billiard orbits if T has smooth boundary.) This algorithm is finite if T is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular n-gon R n is 2-bounce for \({n \ge 4}\), with length twice the width of R n . As an application we obtain an algorithm computing the Ekeland–Hofer–Zehnder capacity of the four-dimensional domain \({T \times B^2}\) in the standard symplectic vector space \({{\mathbb{R}}^4}\). Our method is based on the work of Bezdek–Bezdek (Geom. Dedicata 141:197–206, 2009) and on the uniqueness of the Fagnano triangle in acute triangles. It works, more generally, for planar Minkowski billiards.

Homotopical rigidity of polygonal billiards

Consider two k-gons P and Q. We say that the billiard flows in P and Q are homotopically equivalent if the set of conjugacy classes in the fundamental group of P, viewed as a punctured sphere, which contain a periodic billiard orbit agrees with the analogous set for Q. We study this equivalence relationship and compare it to the notions of order equivalence and code equivalence, introduced in [1,2]. In particular we show if P is a rational polygon, and Q is homotopically equivalent to P, then P and Q are similar, or affinely similar if all sides of P are vertical and horizontal.

On Odd-periodic Orbits in Complex Planar Billiards

The famous conjecture of V. Ya. Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper, we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: (1) triangular orbits; (2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to k-reflective real analytic pseudo-billiards with odd k, the real piecewise-algebraic Ivrii's conjecture and its analogue in the invisibility theory: Plakhov's invisibility conjecture.

Periodic billiard orbits of self-similar Sierpiński carpets

We identify a collection of periodic billiard orbits in a self-similar Sierpiński carpet billiard table Ω(Sa). Based on our refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in Sa, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpiński carpet billiard tables Ω(Sa,n). The trivial limit of this sequence then constitutes a periodic orbit of Ω(Sa). We also determine the corresponding translation surface S(Sa,n) for each prefractal table Ω(Sa,n), and show that the genera {gn}n=0∞ of a sequence of translation surfaces {S(Sa,n)}n=0∞ increase without bound. Various open questions and possible directions for future research are offered.

The existence of a billiard orbit in the regular hyperbolic simplex

In this note we establish the existence of a n + 1 periodic billiard trajectory inside an n-dimensional regular simplex in the hyperbolic space, which hits the interior of every facet exactly once.

Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards

The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult, table to analyze. In this paper, we approach this problem by approximating (from the inside) Omega(KS) by well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We first show that the flat surface S(KS_n) determined from the rational billiard Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [Gut2], allows us to define a sequence of compatible orbits of prefractal billiards. We define a hybrid orbit of a prefractal billiard Omega(KS_n) and show that every dense orbit of a prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits. We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. Finally, we discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining `fractal flat surfaces' naturally associated with the billiard flows.

Hausdorff dimension of three-period orbits in Birkhoff billiards

We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.