Elliptical Billiards Research Articles

Classical and quantum elliptical billiards: Mixed phase space and short-range correlations in singlets and doublets

Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory, these systems are simple prototypes. The conservative dynamics of a billiard may vary from regular to chaotic, depending only on the shape of the boundary. This work aims to shed light into the quantization of classically chaotic systems. We present numerical results on classical and quantum properties in two bi-parametric families of billiards, namely Elliptical Stadium Billiard (ESB) and Elliptical-C3 Billiards (E-C3B). Both families entail elliptical perturbations of chaotic billiards with originally circular sectors on their borders. Our numerical calculations provide evidence that these elliptical families may exhibit a mixed classical dynamics, identified by the chaotic fraction of the phase space, the parameter χc<1. We use this quantity to guide our analysis of quantum spectra. We explore the short-range correlations through nearest neighbor spacing distribution p(s), revealing that in the mixed region of the classical phase space, p(s) is well described by the Berry–Robnik–Brody (BRB) distributions for the ESB. In agreement with the expectation from the so-called ergodic parameter α=tH/tT, the ratio between the Heisenberg time and the classical diffusive-like transport time. Our findings indicate the possibility of quantum dynamical localization when α<1. For the E-C3B family, the eigenstates can be split into singlets and doublets. BRB successfully describes p(s) for singlets as the previous family in the mixed region. However, for doublets, new distributions recently introduced in the literature come into play, providing descriptions for p(s) with a focus on cases where χc=1. We observed that as χc decreases, the p(s) distributions simultaneously deviate from the Gaussian Orthogonal Ensemble (GOE) for singlets, and Gaussian Unitary Ensemble (GUE) for doublets.

Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture

AbstractWe offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in , there are only finitely many directions for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results (which are shown not to hold for general billiards) have their origin in ‘relative’ cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori.

Elliptic Flowers: New Types of Dynamics to Study Classical and Quantum Chaos.

We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong to a new class of dynamical systems called elliptic flowers billiards. Such systems demonstrate a variety of new behaviors which have never been observed or predicted to exist. Elliptic flowers billiards, where a chaotic (non-chaotic) core coexists with the same (chaotic/non-chaotic) type of dynamics in flows were recently constructed. Therefore, all four possible types of coexisting dynamics in the core and tracks are detected. However, it is just the beginning of studies of elliptic flowers billiards, which have already extended the imagination of what may happen in phase spaces of nonlinear systems. We outline some further directions of investigation of elliptic flowers billiards, which may bring new insights into our understanding of classical and quantum dynamics in nonlinear systems.

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.

Integrable elliptic billiards and ballyards

The billiard problem concerns a point particle moving freely in a region of the horizontal plane bounded by a closed curve Γ, and reflected at each impact with Γ. The region is called a ‘billiard’, and the reflections are specular: the angle of reflection equals the angle of incidence. We review the dynamics in the case of an elliptical billiard. In addition to conservation of energy, the quantity L1 L2 is an integral of the motion, where L1 and L2 are the angular momenta about the two foci. We can regularize the billiard problem by approximating the flat-bedded, hard-edged surface by a smooth function. We then obtain solutions that are everywhere continuous and differentiable. We call such a regularized potential a ‘ballyard’. A class of ballyard potentials will be defined that yield systems that are completely integrable. We find a new integral of the motion that corresponds, in the billiards limit, to L1 L2. Just as for the billiard problem, there is a separation of the orbits into boxes and loops. The discriminant that determines the character of the solution is the sign of L1 L2 on the major axis.

Circular and Elliptical Neutrino Billiards: A Semiclassical Approach

Circular and Elliptical Neutrino Billiards: A Semiclassical Approach

Elliptic Billiards and Ellipses Associated to the 3-Periodic Orbits

The centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses. In this article, we obtain the canonical equations of these ellipses. Moreover, we complement the previous results about incenters obtained by Romaskevich. Also the geometric locus defined by barycenters of the triangles of an elliptic billiard are ellipses as established by Schwartz and Tabachnikov and explicit equations of these ellipses are also given. In addition, we obtain that the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse and the canonical equation is obtained.

Checkerboard incircular nets: Laguerre geometry and parametrisation

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties which are also present in the case of checkerboard IC-nets. The parametrisation obtained in this manner is reminiscent of that associated with elliptic billiards. Connections with discrete confocal coordinate systems and the fundamental QRT maps of integrable systems theory are made. The formalism developed in this paper is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.

Elliptical Billiards in the Minkowski Plane and Extremal Polynomials

Elliptical Billiards in the Minkowski Plane and Extremal Polynomials

On the local Birkhoff conjecture for convex billiards

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.

Two-Valued Groups, Kummer Varieties, and Integrable Billiards

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $$\sigma $$ -functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were ”virtual”, while in the present case the resulting bundles are effectively realizable.

On Near Integrability of Some Impact Systems

A class of Hamiltonian impact systems exhibiting smooth near-integrable behavior is presented. The underlying unperturbed model investigated is an integrable, separable, 2 degrees of freedom mechanical impact system with effectively bounded energy level sets and a single straight wall which preserves the separable structure. Singularities in the system appear either as trajectories with tangent impacts or as singularities in the underlying Hamiltonian structure (e.g., separatrices). It is shown that away from these singularities, a small perturbation from the integrable structure results in smooth near-integrable behavior. Such a perturbation may occur from a small deformation or tilt of the wall which breaks the separability upon impact, the addition of a small regular perturbation to the system, or a combination of both. In some simple cases explicit formulas for the leading order term in the near-integrable return map are derived. Near integrability is also shown to persist when the hard billiard boundary is replaced by a singular, smooth, steep potential, thus extending the near-integrability results beyond the scope of regular perturbations. These systems constitute an additional class of examples of near-integrable impact systems, beyond the traditional one dimensional oscillating billiards, nearly elliptic billiards, and the near-integrable behavior near the boundary of convex smooth billiards with or without magnetic field.

Topological Invariants for Elliptical Billiards and Geodesics on Ellipsoids in the Minkowski Space

We describe topological properties of the elliptical billiard in the Minkowski plane and geodesic motion on an ellipsoid in the Minkowski space.

On semidiscrete constant mean curvature surfaces and their associated families

The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete sinh -Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.

On the incenters of triangular orbits on elliptic billiards

We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have suggested that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection.

Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism.

Some dynamical properties for an oval billiard with a scatterer in its interior are studied. The dynamics consists of a classical particle colliding between an inner circle and an external boundary given by an oval, elliptical, or circle shapes, exploring for the first time some natural generalizations. The billiard is indeed a generalization of the annular billiard, which is of strong interest for understanding marginally unstable periodic orbits and their role in the boundary between regular and chaotic regions in both classical and quantum (including experimental) systems. For the oval billiard, which has a mixed phase space, the presence of an obstacle is an interesting addition. We demonstrate, with details, how to obtain the equations of the mapping, and the changes in the phase space are discussed. We study the linear stability of some fixed points and show both analytically and numerically the occurrence of direct and inverse parabolic bifurcations. Lyapunov exponents and generalized bifurcation diagrams are obtained. Moreover, histograms of the number of successive iterations for orbits that stay in a cusp are studied. These histograms are shown to be scaling invariant when changing the radius of the scatterer, and they have a power law slope around -3. The results here can be generalized to other kinds of external boundaries.

Computing Mather's $\beta$-function for Birkhoff billiards

This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum . After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.

Neutral particle focusing in composite driven dissipative billiards

Dynamical focusing of ensembles of neutral particles in energy and configuration space has been demonstrated recently (Petri et al. in Phys. Rev. E (R) 82:035204, 2010) using time-dependent elliptical billiards. The interplay of nonlinearity, dissipation, and driving yields the occurrence of attractors in the phase space of the billiard. Here, we show that dissipative oval billiards with slowly oscillating elliptical scatterers in the interior allow for a dynamical focusing on simple periodic trajectories with close to perfect efficiency. This setup should be more amenable to corresponding experiments of certain type which are briefly discussed.

Analysis of resonant population transfer in time-dependent elliptical quantum billiards

A Fermi golden rule for population transfer between instantaneous eigenstates of elliptical quantum billiards with oscillating boundaries is derived. Thereby the occurrence of both the recently observed resonant population transfer between instantaneous eigenstates and the empirical criterion stating that these transitions occur when the driving frequency matches the mean difference of the latter [Lenz et al., New J. Phys. 13, 103019 (2011)] is explained. As a second main result a criterion judging which resonances are resolvable in a corresponding experiment of certain duration is provided. Our analysis is complemented by numerical simulations for three different driving laws. The corresponding resonance spectra are in agreement with the predictions of both criteria.

Non-persistence of resonant caustics in perturbed elliptic billiards

AbstractA caustic of a billiard table is a curve such that any billiard trajectory, once tangent to the curve, stays tangent after every reflection at the boundary. When the billiard table is an ellipse, any non-singular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics—those 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.