Billiard Motion Research Articles

Rotating rod and ball

We consider a mechanical system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points. The ball is reflected elastically when colliding with the rod and moves freely between consecutive hits. A sliding motion along the rod is also allowed. We prove the existence and uniqueness of the motion with a given position and velocity at a certain time instant. We prove that only 5 kinds of motion are possible: a billiard motion; a sliding motion; a billiard motion followed by sliding; a sliding motion followed by a billiard one; and a constant motion when the ball is at the center of rotation. The asymptotic behaviors of time intervals between consecutive hits and of distances between the points of hits on the rod are determined.

The geometry of billiards in ellipses and their poncelet grids

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They define the associated Poncelet grid. If a billiard is periodic then it closes for any choice of the initial vertex on the ellipse. This gives rise to a continuous variation of billiards which is called billiard motion though it is neither a Euclidean nor a projective motion. The extension of this motion to the associated Poncelet grid leads to new insights and invariants.

Classification of billiard motions in domains bounded by confocal parabolas

We consider the billiard dynamical system in a domain bounded by confocal parabolas. We describe such domains in which the billiard problem can be correctly stated. In each such domain we prove the integrability for the system, analyse the arising Liouville foliation, and calculate the invariant of Liouville equivalence — the so-called marked molecule. It turns out that billiard systems in certain parabolic domains have the same closures of solutions (integral trajectories) as the systems of Goryachev-Chaplygin-Sretenskii and Joukowski at suitable energy levels. We also describe the billiard motion in noncompact domains bounded by confocal parabolas, namely, we describe the topology of the Liouville foliation in terms of rough molecules. Bibliography: 16 titles.

Relativistic wavepackets in classically chaotic quantum cosmological billiards

Close to a spacelike singularity, pure gravity and supergravity in 4 to 11 spacetime dimensions admit a cosmological billiard description based on hyperbolic Kac-Moody groups. We investigate the quantum cosmological billiards of relativistic wavepackets towards the singularity, employing flat and hyperbolic space descriptions for the quantum billiards. We find that the strongly chaotic classical billiard motion of four-dimensional pure gravity corresponds to a spreading wavepacket subject to successive redshifts and tending to zero as the singularity is approached. We discuss the possible implications of these results in the context of singularity resolution and compare them with those of known semiclassical approaches. As an aside, we obtain exact solutions for the one-dimensional relativistic quantum billiards with moving walls.

The Frequency Map for Billiards inside Ellipsoids

The billiard motion inside an ellipsoid $Q\subset\mathbb{R}^{n+1}$ is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension n. The motion on each Liouville torus becomes just a parallel translation with some frequency $\omega$ that varies with the torus. Further, any billiard trajectory inside Q is tangent to n caustics $Q_{\lambda_1},\ldots,Q_{\lambda_n}$, so the caustic parameters $\lambda=(\lambda_1,\ldots,\lambda_n)$ are integrals of the billiard map. The frequency map $\lambda\mapsto\omega$ is a key tool for understanding the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present two conjectures, fully supported by numerical experiments. We obtain, from one of the conjectures, some lower bounds on the periods. These bounds depend only on the type of the n caustics. We describe the geometric meaning, domain, and range of $\omega$. The map $\omega$ can be continuously extended to singular values of the caustic parameters, although it becomes “exponentially sharp” at some of them. Finally, we study triaxial ellipsoids of $\mathbb{R}^3$. We numerically compute the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of $\mathbb{R}^2$.

Spacelike Singularities and Hidden Symmetries of Gravity.

We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

COSMOLOGICAL SINGULARITIES, EINSTEIN BILLIARDS AND LORENTZIAN KAC–MOODY ALGEBRAS

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii–Khalatnikov–Lifshitz and technically simplified by the use of the Arnowitt–Deser–Misner Hamiltonian formalism) that the asymptotic behavior, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String Theory, the billiard tables describing asymptotic cosmological behavior are found to be identical to the Weyl chambers of some Lorentzian Kac–Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac–Moody group E10, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10.

Persistence of homoclinic orbits for billiards and twist maps

We consider the billiard motion inside a C2-small perturbation of an n-dimensional ellipsoid Q with a unique major axis. The diameter of the ellipsoid Q is a hyperbolic two-periodic trajectory whose stable and unstable invariant manifolds are doubled, so that there is an n-dimensional invariant set W of homoclinic orbits for the unperturbed billiard map. The set W is a stratified set with a complicated structure.For the perturbed billiard map the set W generically breaks down into isolated homoclinic orbits. We provide lower bounds for the number of primary homoclinic orbits of the perturbed billiard which are close to unperturbed homoclinic orbits in certain strata of W.The lower bound for the number of persisting primary homoclinic billiard orbits is deduced from a more general lower bound for exact perturbations of twist maps possessing a manifold of homoclinic orbits.

Periodic orbit spectrum in terms of Ruelle-Pollicott resonances.

Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g., a trajectory "p" returns to its initial conditions after some fixed time tau(p). Our aim is to investigate the spectrum [tau(1),tau(2), ...] of periods of the periodic orbits. An explicit formula for the density rho(tau)= Sigma(p)delta(tau-tau(p)) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle-Pollicott resonances). For large periods, corrections to the well-known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random-matrix theory, and discrete maps are also considered. In particular, a random-matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.

Homoclinic billiard orbitsinside symmetrically perturbed ellipsoids

The billiard motion inside an ellipsoid ofℝ3 is completely integrable. Ifthe ellipsoid is not of revolution, thereare many orbits bi-asymptotic to its majoraxis. The set of bi-asymptotic orbits isdescribed from a geometrical, dynamicaland topological point of view. It containseight surfaces, called separatrices.The splitting of the separatrices under symmetric perturbations of theellipsoid is studied using a symplectic discrete version of thePoincaré-Melnikov method, with special emphasis on the followingsituations:close to the flat limit(when the minor axis of the ellipsoid is small enough), close to the oblatelimit(when the ellipsoid is close to an ellipsoid of revolution around its minoraxis)and close to the prolate limit(when the ellipsoid is close to an ellipsoid of revolution around its majoraxis).It is proved that any non-quadratic entire symmetric perturbation breaksthe integrability and splits the separatrices, although (at least) 16symmetric homoclinic orbits persist. Close to the flat limit, these orbitsbecome transverse under very general polynomial perturbations of theellipsoid.Finally, a particular quartic symmetric perturbation is analysed in greatdetail. Close to the flat and to the oblate limits, the 16 symmetrichomoclinic orbits are the unique primary homoclinic orbits. Close to theprolate limit, the number of primary homoclinic orbits undergoesinfinitely many bifurcations. The first bifurcation curves are computednumerically.The planar and high-dimensional cases are also discussed.

E10, BE10 and arithmetical chaos in superstring cosmology.

It is shown that the neverending oscillatory behavior of the generic solution, near a cosmological singularity, of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in nine-dimensional hyperbolic space. The Coxeter group of reflections of this billiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algebra E10 (for type II) or BE10 (for type I or heterotic), which are both arithmetic. These results lead to a proof of the chaotic ("Anosov") nature of the classical cosmological oscillations, and suggest a "chaotic quantum billiard" scenario of vacuum selection in string theory.

Morphological image analysis of quantum motion in billiards.

Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two- and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability distribution provides a well defined quantity to distinguish between classically integrable and nonintegrable billiards. In three dimensions the time-averaged mean breadth of the probability distribution may also be used for this purpose.

Stochastic transition of free motion in billiards with borders given by equipotential lines of the diamagnetic Kepler problem.

We compare the transition from ordered to chaotic motion in a potential with that in a billiard, whose borders are defined by the maximal equipotential line of the potential system in the case of hydrogen in a uniform magnetic field. For the billiard we calculate the Poincar\'e maps, the fraction of regular motion on the surface of section, and the stability properties of the shortest periodic orbits. In contrast to the H atom, the billiard shows a generic transition to chaos. While the shape of the orbits is determined by the boundary and is thus very similar, their properties of stability are different: The potential tends to stabilize the motion. The onset of instability in the billiard can be understood in terms of the curvature of the boundary; for the potential system the Gaussian curvature of the potential-energy surface is shown to be the relevant parameter.

Chaos in a periodic three-particle system under Yukawa interaction.

The dynamics of a three-particle motion under singular potentials such as Yukawa and Coulomb provides a Hamiltonian system leading to the billiard system in a high-energy region, which clarifies the interrelation between these two systems. From numerical analyses of phase pictures, trajectories in a real space, winding numbers, and Farey trees, we find several interesting facts: a strong similarity on the topological structure of chaos irrespective of the exchange of the nature of fixed points; a close relation between the symmetry and the winding numbers, i.e., the ${C}_{3v}$ completely symmetric orbit leads to the coprime rational and the partially symmetric one to the noncoprime one; a classification of the symmetry of orbits on the Farey tree. The results on the symmetry and winding number shed light on the question of the integrability. On the other hand, in a low-energy region, the dynamics is described by the H\'enon-Heiles system and hence the billiard motion contains both features of the ${C}_{3v}$ symmetry and a circle.

Numerical study of billiard motion in an annulus bounded by non-concentric circles

This paper exposes a simple, focusing-dispersing billiard system whose phase space simultaneously exhibits the full range of possible Hamiltonian syste, orbit behavior over a continuous range of system parameter values. Specifically, we find that billiard motion between two non-concentric circles is characterized by three distinct phase space regions: (1) a rigorously integrable region in which billiard orbits undergo collisions only with the outer circle boundary; (2) a KAM near-integrable region in which billiard collisions strictly alternate between inner and outer circles; and (3) a chaotic region produced by orbital sequences of billiard collisions which randomly alternate between the integrable and near-integrable patterns. The properties of the integrable, near-integrable, and chaotic regions, the presence of island chains, and the homoclinic points associated with certain hyperbolic fixed points are discussed. Perhaps the most interesting feature of this billiard system, however, is the fresh view of the source of chaos it provides; specifically, the chaotic orbits of region (3) exhibit “random” jumping between the extended invariant curves of regions (1) and (2).