Billiard Dynamics Research Articles

Динамика биллиардa: обзорная статья с акцентом на нерешенные задачи

Динамика биллиардa: обзорная статья с акцентом на нерешенные задачи

A theoretical analysis of billiard ball dynamics under cushion impacts

The last two decades have seen a growing interest in research related to billiards. There have been a number of projects aimed at developing training systems, robots, and computer simulations for billiards. Determination of billiard ball trajectories is important for all of these systems. The ball's collision with a cushion is often encountered in billiards and it drastically changes the ball trajectory, especially when the ball has spin. This work predicts ball bounce angles and bounce speeds for the ball's collision with a cushion, under the assumption of insignifi-cant cushion deformation. Differential equations are derived for the ball dynamics during the impact and these equations are solved numerically. The numerical solutions together with previous experimental work by the authors predict that for the ball—cushion collision, the values of the coefficient of restitution and the sliding coefficient of friction are 0.98 and 0.14, respectively. A comparison of the numerical and experimental results indicates that the limiting normal velo-city under which the rigid cushion assumption is valid is 2.5 m/s. A number of plots that show the rebound characteristics for given ball velocity—spin conditions are also provided. The plots quantify various phenomena that have hitherto only been described in the billiards literature.

Recurrence for quenched random Lorentz tubes

We consider the billiard dynamics in a striplike set that is tessellated by countably many translated copies of the same polygon. A random configuration of semidispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global choice of scatterers, is called quenched random Lorentz tube. We prove that under general conditions, almost every system in the ensemble is recurrent.

Chaotic dynamics in classical nuclear billiards

We consider several noninteracting nucleons moving in a 2D Woods–Saxon type potential well and hitting the vibrating surface. The Hamiltonian has a coupling term between the particle motion and the collective coordinate which generates a self-consistent dynamics. The numerical simulation is based on the solutions of the Hamilton equations which was solved using an algorithm of Runge–Kutta type (order 4–5) having an optimized step size, taking into account that the absolute error for each variable is less than 10 −6. Total energy is conserved with high accuracy, i.e., approx. 10 −6 in absolute value. We analyze the chaotic behavior of the nonlinear dynamics system using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropies. A qualitative and quantitative picture of the achievement of soft chaos is shown for a comparative study between the adiabatic and the resonance stage of nuclear interaction. We consider that the onset of chaos would be linked to the resonance stage of interaction. This assumption is argued in [1].

Indication for quantum Darwinism in electron billiards

In this paper, we investigate the dynamics in electron billiards by using classical and quantum mechanical calculations. We report on the existence of pointer states in single-dot and double-dot electron billiards. Additionally, we show that the two types of pointer states have the propensity to create offspring, i.e. they can be observed in the individual modes propagating between the external reservoirs. This can be understood as an indication that quantum Darwinism is present in the electron billiards.

Application of high-speed imaging to determine the dynamics of billiards

In spite of interest in the dynamics of the billiards family of games (for example, pool and snooker), experiments using present-day inexpensive and easily accessible cameras have not been reported. We use a single high-speed camera and image processing techniques to track the trajectory of snooker balls to 1 mm accuracy. Successive ball positions are used to measure the dynamical parameters involved in snooker. Values for the rolling and the sliding coefficients of friction were found. The cushion-ball impact was studied for impacts perpendicular to the cushion. The separation angles and separation velocities after an oblique collision were measured and compared with predicted values. Our measurement technique is a simple, reliable, fast, and nonintrusive method, which can be used to test the numerous theories for the dynamics of billiards. The addition of a spin tracking element would further broaden its capabilities.

Fermionic Kac-Moody billiards and supergravity

We study the billiards, i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit of eleven-dimensional supergravity and of its dimensional truncations (notably four-dimensional simple supergravity). By exploiting the gravity-coset correspondence, we show that the billiard dynamics of the gravitino is described by a `spin extension' of the E(10) Weyl group, namely as a product of 90 degree `vector-spinor rotations' along certain simple-root-related generators of the maximal compact subalgebra K(E(10)) of the hyperbolic Kac--Moody algebra E(10). The `super-billiard' that combines the bosonic and fermionic billiards is found to have a remarkably simple structure, which exhibits a striking analogy with a polarized photon propagating in the ten-dimensional Lorentzian Weyl chamber of E(10).

Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: Beyond Poncelet porisms

The thirty years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper. Starting with the observation of the billiard nature of some classical constructions and configurations, we construct the billiard algebra, that is a group structure on the set T of lines simultaneously tangent to d − 1 quadrics from a given confocal family in the d-dimensional Euclidean space. Using this tool, the related results of Reid, Donagi and Knörrer are further developed, realized and simplified. We derive a fundamental property of T: any two lines from this set can be obtained from each other by at most d − 1 billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: s-skew lines in T and s-weak Poncelet trajectories, s = − 1 , 0 , … , d − 2 . The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in this paper enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results: Cayley's theorem, Weyr's theorem, Griffiths–Harris theorem and Darboux theorem.

Prevalence of marginally unstable periodic orbits in chaotic billiards

The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and stands apart from the regular regions. We show that these structures both exist and strongly influence the dynamics of locally perturbed billiards, which include a large class of widely studied systems. We demonstrate the impact of these structures in the quantum regime using microwave experiments in annular billiards.

Tits–Satake projections of homogeneous special geometries

We organize the homogeneous special geometries, describing as well the couplings of D = 6, 5, 4 and 3 supergravities with eight supercharges, in a small number of universality classes. This relates manifolds on which similar types of dynamical solutions can exist. The mathematical ingredient is the Tits–Satake projection of real simple Lie algebras, which we extend to all solvable Lie algebras occurring in these homogeneous special geometries. Apart from some exotic cases all the other, ‘very special’, homogeneous manifolds can be grouped into seven universality classes. The organization of these classes, which capture the essential features of their basic dynamics, commutes with the r- and c-map. Different members are distinguished by different choices of the paint group, a notion discovered in the context of cosmic billiard dynamics of non-maximally supersymmetric supergravities. We comment on the usefulness of this organization in universality class both in relation with cosmic billiard dynamics and with configurations of branes and orbifolds defining special geometry backgrounds.

On Chaotic Dynamics in Rational Polygonal Billiards

We discuss the interplay between the piece-line regular and vertex-angle singular boundary eects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular motion in polygons is taken within the alternative deterministic and stochastic frameworks. The analysis is developed in terms of the billiard-wall collision distribution and the particle survival probability, sim- ulated in closed and weakly open polygons, respectively. In the multi-vertex polygons, the late-time wall-collision events result in the circular-like regular periodic trajectories (sliding orbits), which, in the open billiard case are likely transformed into the surviving collective excitations (vortices). Having no topological analogy with the regular orbits in the geomet- rically corresponding circular billiard, sliding orbits and vortices are well distinguished in the weakly open polygons via the universal and non-universal relaxation dynamics. process caused by the boundary, formed by the disk and the square, and the interplay between boundary segments, formed by the semi-circles and the square. Moreover, rational polygons of m equal sides and m equal vertices (hereafter, m-gons (1)) exhibit chaotic-like changes in the associated quantum-level spectra (4), but the fluctuations found (5) are very close to those known in the universal Gaussian statistics. These controversial evidences for the chaotic and regular behavior of polygons are a challenge to dynamic theory of billiards. In the present paper we develop a physical insight into the problem, based on analysis of the simulation data on the billiard collision statistics. Within a more general context, the delicate problem of the interplay between regular and irregular segments, which constitute the total billiard boundary, is ultimately related to apparent controversy between the causality and randomness. Our approach to the problem is developed through the alternative deterministic and stochastic frameworks, which elucidate the duality of chaotic and non-chaotic features in the intrinsic dynamics of the rational polygons. The singular eects,

Lyapunov spectra of billiards with cylindrical scatterers: Comparison with many-particle systems

The dynamics of a system consisting of many spherical hard particles can be described as a single point particle moving in a high-dimensional space with fixed hypercylindrical scatterers with specific orientations and positions. In this paper, the similarities in the Lyapunov exponents are investigated between systems of many particles and high-dimensional billiards. The billiards contain cylindrical scatterers which have isotropically distributed orientations and homogeneously distributed positions. The dynamics of the isotropic billiards are calculated using a Monte Carlo simulation, and a reorthogonalization process is used to find the Lyapunov exponents. The results are compared to numerical results for systems of many hard particles as well as the analytical results for the high-dimensional Lorentz gas. The smallest three-quarters of the positive exponents behave more like the exponents of hard-disk systems than the exponents of the Lorentz gas. This similarity shows that the hard-disk systems may be approximated by a spatially homogeneous and isotropic system of scatterers for a calculation of the smaller Lyapunov exponents, apart from the exponent associated with localization. The method of the partial stretching factor is used to calculate these exponents analytically, with results that compare well with simulation results of hard disks and hard spheres.

Switched flow systems: pseudo billiard dynamics

We study a class of dynamical systems which generalizes and unifies some models arising in the analysis of switched flow systems in manufacturing. The general properties of these dynamical systems, called pseudo billiards, as well as some of their perturbations are discussed.

Level dynamics in pseudointegrable billiards: an experimental study

The level dynamics of pseudointegrable systems with different genus numbers g is studied experimentally using microwave cavities. For higher energies, the distribution of the eigenvalue velocities is Gaussian, as it is expected for chaotic systems with time-reversal symmetry and shows no dependence on g. Also the curvature distribution P( k) for large k is decaying as it is expected for chaotic systems, i.e., P( k)∼| k| −3. For small k an intermediate behavior is found, where P( k) changes from integrable towards chaotic behavior with growing g.

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

Geometrical nonlinear dynamics features of systems with elastic reflections

Within the framework оf the new geometrical-dynamic approach а special class оf dynamic systems - reversible mappings with projective involutions оп «symmetric» phase space - is linked with billiard systems. Basic geometrical features оf locally smooth billiard involutions - projectivity and piecewise discontinuity are explored and their role in making оf one оr another (regular and random) nonlinear billiard dynamics is indicated. Billiard involutions for simple algebraic curves are obtained and their common properties are set.

Dynamics of billiards with periodically time-dependent boundaries

Dynamical systems corresponding to billiards with periodic perturbed boundaries are investigated. A general approach going back to the Aubrey — Mather theory is proposed. On the basis of investigations of the billiard maps, planar billiards with time— dependent boundaries in the form of stadium and circle, and billiards оп two—dimensional sphere with the boundary formed by a disk of the varying radius are considered.

Billiards in Finsler and Minkowski geometries

We begin the study of billiard dynamics in Finsler geometry. We deduce the Finsler billiard reflection law from the “least action principle”, and extend the basic properties of Riemannian and Euclidean billiards to the Finsler and Minkowski settings, respectively. We prove that the Finsler billiard map is a symplectomorphism, and compute the mean free path of the Finsler billiard ball. For the planar Minkowski billiard we obtain the mirror equation, and extend the Mather’s non-existence of caustics result. We establish an orbit-to-orbit duality for Minkowski billiards.

Observation of chaotic and regular dynamics in atom-optics billiards.

We report on experimental observations of chaotic and regular motion of ultracold atoms confined by a billiard-shaped optical dipole potential induced by a rapidly scanning laser beam. To investigate the dynamics of the atoms confined by such an "atom-optics" billiard we measure the decay of the number of trapped atoms through a hole on the boundary. A fast and purely exponential decay, the clear signature of chaotic motion, is found for a stadium billiard, but not for a circular or an elliptical billiard, in agreement with theory. We also investigated the effects of decoherence, velocity spread, and gravity on regular and chaotic motion.

Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of the billiard dynamics, and give lower estimation for the Lyapunov exponent. This extends our recent results for non-magnetic billiards on surfaces of constant curvature. Using these conditions, we construct large classes of magnetic billiard tables with positive Lyapunov exponents on the plane, on the sphere and on the hyperbolic plane.