Dynamical Billiards

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Curator: Leonid Bunimovich

Figure 1: Sinai billiards

Dynamical Billiard is a dynamical system corresponding to the inertial motion of a point mass within a region \(\Omega\) that has a piecewise smooth boundary with elastic reflections. The angle of reflection equals the angle of incidence from the boundary.

Billiards appear as natural models in many problems of optics, acoustics and classical mechanics. The most prominent model of statistical mechanics, the Boltzmann gas of elastically colliding hard balls in a box can be easily reduced to a billiard. In fact, it was the still unproved Boltzmann-Sinai hypothesis on ergodicity of the gas of elastically interacting hard balls in a torus which essentially stimulated a development of the theory of billiards. A general feeling though is that a complete proof should be coming soon (Simanyi, 2003).


General Properties

The orbits of billiards are broken lines in configuration space \(\Omega\) with the segments corresponding to the free paths within the region and the vertices corresponding to the reflections off its boundary. The region \(\Omega\) is also called a billiard table. If \(\Omega\) is a region on a Riemannian manifold then the orbits consist of geodesic (rather than straight) segments. Billiard is a dynamical system with continuous time. However, dynamics of billiards can be rather completely characterized by a billiard map which transforms coordinates and incident angle of the point of reflection into the coordinates and the incident angle at the point of the next reflection from the boundary. Billiards models are Hamiltonian systems with potential \(V(q)\) that is equal to zero within a billiard table \(\Omega\) and infinity outside \(\Omega\ .\) Hence, the phase volume is preserved under the dynamics and in many cases one can neglect such sets of orbits which have phase volume zero. In particular, the set of all orbits which hit singular points of the boundary of a billiard table has phase volume zero, and therefore billiard dynamics is well defined on a subset of the phase space which has a full phase volume.

The dynamics of billiards is completely defined by the shape of its boundary and it demonstrates all the variety of possible behaviors of Hamiltonian systems from integrable to completely chaotic ones. A smooth component of the boundary is dispersing, focusing or neutral if it is convex inward, outward the billiard region, or if it is flat (has zero curvature) respectively.

Billiards in Polygons and Polyhedra

Let a billiard table \(\Omega\) have a flat component \(\Gamma\) of the boundary \(\partial\Omega\ .\) Reflect \(\Omega\) with respect to \(\Gamma\) and consider a "double" billiard table \(\Omega^*\) which is the union of \(\Omega\) and its reflected copy. The orbits of the billiard in \(\Omega^*\) are "straightened" in the sense that to each two consecutive links of an orbit of a billiard in \(\Omega\) which are separated by a reflection from \(\Gamma\) corresponds a single link of the corresponding billiard orbit in \(\Omega^*\ .\) In case of billiards in polygons and polyhedra all smooth components of the boundary \(\partial\Omega\) are flat and therefore one can turn the billiard orbits into straight lines by consecutive reflections of the billiard table with respect to its faces (sides) at each reflection. Therefore periodic orbits in billiards in polygons are never isolated. A polygon is called rational if all its angles are commensurable with \(\pi\ .\) Each billiard orbit in a rational polygon can have only a finite number of directions and also there exist only a finite number of copies of \(\Omega\) obtained by the reflections with respect to its sides. Therefore, one can construct a Riemann surface from a finite number of copies of \(\Omega\) and consider there a directional flow. This flow has a conserved quantity which allows the application of the powerful techniques from Teichmuller theory. In particular, on almost all invariant manifolds the system is uniquely ergodic, i.e., has an unique invariant measure. A lot of impressive results about a fine structure of billiard orbits in rational polygons were recently obtained (see the review paper Masur, Tabachnikov, 2002).

Billiards in rational polygons are nonergodic because of a finite number of possible directions of their orbits. However, a billiard in a typical polygon is ergodic (Kerckhoff et al., 1986). Nevertheless, billiards in polygons and polyhedra have zero metric (Kolmogorov-Sinai) entropy (Boldrighini, et al., 1978) and therefore all these billiards are nonchaotic. If the sequence of the faces (sides) of \(\Omega\) from which an orbit has reflections in the past is known then it uniquely defines the sequence of faces from which it will be reflected in the future.

Billiards in polygons and polyhedra appear in some systems of classical mechanics. For instance, a system of \(N\) point masses moving freely in a segment and elastically colliding with its ends and between themselves can be reduced to the billiard in a \(N\)-dimensional polyhedron.

Integrable and Smooth Convex Billiards

Classical examples of integrable billiards are billiards in circles and ellipses. The boundary of these billiard tables consists of one smooth focusing component. Configuration spaces of these billiards are foliated by caustics, which are smooth curves (or surfaces) such that if one segment of the billiard orbit is tangent to it, then every other segment of this orbit is also tangent to it. Billiards in a circle has one family of caustics formed by (smaller) concentric circles, while billiards in an ellipse has two families of caustics (confocal ellipses and confocal hyperbolas). Trajectories tangent to different families of caustics are separated by orbits such that each segment intersects a focus of the ellipse. It was conjectured by Birkhoff (Birkhoff, 1927) that among all billiards inside smooth convex curves, only billiards in ellipses are integrable. In the dimensions greater than two though only in billiards in ellipsoids a billiard table is foliated into smooth convex caustics (Berger, 1995). It does not imply, however, that only billiards in ellipsoids are integrable because if a billiard in dimension greater than two has an invariant hypersurface then this hypersurface does not necessarily consist of rays tangent to some hypersurface in the configuration space. If a boundary of a two-dimensional billiard table is strictly convex, sufficiently smooth and its curvature never vanishes, then there exists an uncountable number of smooth caustics in the vicinity of the boundary, and moreover, the phase volume of the orbits tangent to these caustics is positive (Lazutkin, 1973). However, if the curvature of the boundary vanishes at some point, then there are no caustics in the vicinity of the boundary (Mather, 1984).

Birkhoff (1927) proved that for every integer \(n\ge 2\) and every \(r\le n/2\ ,\) coprime with \(n\ ,\) there exist at least two \(n\)-periodic billiard trajectories making \(r\) full rotations each period. If \(\Omega\) is a smooth strictly convex closed billiard table in \(d\)-dimensional Euclidean space then generically a number of \(n\)-periodic billiard orbits in \(\Omega\) is not less than \((d-1)(n-1)\) (Farber, Tabachnikov, 2002).

Chaotic Billiards

Most classes of billiards demonstrate chaotic behavior. A reason is that a typical billiard table \(\Omega\) has at least one nonflat component of the boundary and that essentially influences the dynamics of the corresponding billiard. A general belief is that a typical billiard is chaotic, i.e., it has a positive Kolmogorov-Sinai entropy. Therefore a key question in the theory of billiards is concerned with the mechanisms of chaos in these systems.

Dispersing and Semi-Dispersing Billiards

Billiards with the strongest chaotic properties have the boundary which is everywhere dispersing. These billiards were introduced by Sinai in his seminal paper (Sinai, 1970) which laid a foundation for analysis of ergodic and statistical properties of hyperbolic dynamical systems with singularities. In billiards singularities appear because of tangencies of orbits with the boundary of a billiard region and because of singularities of this boundary. Billiards with everywhere dispersing boundary are called dispersing billiards. Dispersing billiards with smooth boundary are called Sinai billiards (Figure 1).

Sinai billiards have the strongest possible chaotic properties, being ergodic, mixing, Bernoulli, having a positive Kolmogorov-Sinai entropy and an exponential decay of correlations. All these properties are ensured by one of the fundamental mechanisms of chaos (of hyperbolicity) which is called the mechanism of dispersing. If a parallel beam of rays is fallen onto a dispersing boundary then after reflection it becomes divergent. Therefore the distance between the rays in this beam increases with time and this process of divergence continues after any reflection of this beam from dispersing boundary.

Dispersing boundary plays the same role for billiards as negative curvature does for geodesic flows causing the exponential instability of the dynamics. The Boltzmann gas of hard balls gets reduced to a billiard with the boundary consisting of (intersecting) cylinders. Therefore it has just semi-dispersing boundary and belongs to the class of semi-dispersing billiards. One of the basic questions in the theory of gases of elastic hard balls moving freely in a manifold of infinite volume is an estimate of a maximal number of collisions that may occur before these balls move apart. This estimate for a system of \(N\) hard elastic balls of arbitrary masses and radii moving freely in a simply connected Riemannian space of non-positive sectional curvature reads \(\left( 400 N^2 \frac{m_{\max}}{m_{\min}}\right)^{2N^4} \ ,\) where \(m_{\max}\) and \(m_{\min}\) are the maximal and the minimal masses respectively (Burago, Ferleger, Kononenko, 2000).

Focusing Billiards

Focusing billiards can have the most regular dynamics being integrable ones. Upon the reflection from a focusing boundary a parallel beam of rays becomes convergent, i.e., the result of reflection from a focusing boundary is opposite to the one after reflection from a dispersing boundary. Indeed the distance between the rays in a parallel beam decreases after reflection from a focusing boundary. Although these two processes of focusing and dispersing compete, there exist chaotic billiards in regions having both dispersing and focusing (and possibly neutral as well) components of the boundary (Bunimovich, 1974a). Moreover, a closer analysis of these billiards revealed a new mechanism of chaotic behavior of conservative dynamical systems (Bunimovich, 1974b), which is called a mechanism of defocusing. The key observation is that a narrow parallel beam of rays, after focusing because of reflection from a focusing boundary, may pass a focusing (in linear approximation) point and become divergent provided that a free path between two consecutive reflections from the boundary is long enough. The mechanism of defocusing works under condition that divergence prevails over convergence. From a general point of view the mechanism of dispersing can be viewed as a special case of the mechanism of defocusing when the focusing part of a free path is just absent. Due to this mechanism there exist e.g., focusing billiards with chaotic dynamics. The most famous (although not the first one) among chaotic focusing billiards is a stadium (Figure 2).

Figure 2: Stadium

One obtains a stadium by cutting a circle into two semi-circles and connecting them by two common tangent segments. The length of these segments could be arbitrarily small. Thus the mechanism of defocusing can work under small deformations of even the integrable billiards. Focusing billiards can have as strong chaotic properties as the dispersing billiards do (Bunimovich, 2000; Chernov & Markavian, 2006). Apparently there is no other mechanism of chaos in billiards besides dispersing and defocusing because the flat boundary cannot generate chaotic dynamics.

Because focusing components can belong to the boundary of integrable as well as of chaotic billiards, one may wonder whether there are some restrictions (conditions) that determine (separate) the corresponding types of focusing components. Besides the arcs of the circles two classes of focusing components admissible in chaotic billiards were found (Wojtkowski, 1986; Markarian, 1988). These two classes are, in a sense, dual to each other (Bunimovich, 2000). A general class of focusing components admissible in chaotic billiards is formed by absolutely focusing mirrors (Bunimovich, 1992). Absolutely focusing mirrors form a new notion in geometric optics. A smooth component (or a mirror) \(\gamma\) of a billiard's table boundary is called absolutely focusing if any narrow parallel beam of rays that falls on \(\gamma\) becomes focused after its last reflection in a series of consecutive reflections from \(\gamma\ .\) The notion of absolutely focusing mirrors should be compared with a standard one of (just) focusing mirrors, where a mirror is called focusing if any parallel beam of rays becomes focused just after the first reflection form this mirror. Absolutely focusing mirrors \(\gamma\) can be characterized in terms of their local properties (Donnay, 1991; Bunimovich, 1992) which say that any narrow parallel beam of rays that falls on \(\gamma\) becomes focused after any reflection in a series of consecutive reflections from \(\gamma\ .\)

Clearly, the mechanism of dispersing works in higher than two dimensions as well (Sinai, 1970). However, in dimensions \(d>2\) there is a natural obstacle to the mechanism of defocusing. This obstacle is a phenomenon of astigmatism, according to which the strength of focusing varies in different hyperplanes, and besides in some hyperplanes it could be arbitrarily weak. Nevertheless, the mechanism of defocusing works in higher dimensions as well and chaotic focusing billiards do exist in dimensions \(d\ge 3\) (Bunimovich & Rehacek, 1998). However, one pays a price to astigmatism by not allowing the focusing components of chaotic billiards to be large in \(d>2\) while in \(d=2\) they could be e.g., arbitrarily close to the entire circle.

Billiards with Coexistence of Chaotic and Regular Dynamics

Figure 3: Mushroom

It is a general belief that the phase spaces of typical Hamiltonian systems are divided into KAM-islands and chaotic sea(s). Therefore they are neither integrable nor chaotic ones. The visual and rigorously studied examples of such Hamiltonian systems with divided phase space are billiards in mushrooms (Bunimovich, 2001). The simplest mushroom consist of a semicircular cap sitting on a rectangular stem (Figure 3). A billiard in such mushroom has one integrable island formed by the trajectories which never leave the cap and it is chaotic and ergodic on its complement.

A mushroom becomes a semi-stadium when the width of the feet equals the width of the hat. Combining mushrooms together one gets examples of billiards with an arbitrary (finite or infinite) number of islands coexisting with an arbitrary (finite or infinite) number of chaotic components (Bunimovich, 2001).

Quantum Billiards

Many properties of classical dynamics of billiards are closely related to the properties of the corresponding quantum problem. Consider the stationary Schrodinger equation \(H\psi=E\psi\) with a potential equal zero inside the billiard table \(\Omega\) and equal to infinity on the boundary. Whenever the classical billiards are integrable then the corresponding quantum systems are completely solvable. If a billiard has a smooth convex caustics then there exists an infinite series of eigenfunctions localized in the vicinity of this caustic (Lazutkin, 1991). On the contrary, in (classical) ergodic billiards the eigenfunctions become asymptotically uniformly distributed over the billiard table as the wave numbers tend to infinity (Shnirelman,1991).

Generalized Billiards

Generalized billiards (GB) describe a motion of a mass point (a particle) inside a closed domain \(\Pi \in \mathbb{R}^n\) with the piece-wise smooth boundary \(\Gamma\ .\) On the boundary \(\Gamma\) the velocity of point is transformed as the particle underwent the action of generalized billiard law. GB were introduced by Lev D. Pustyl'nikov in the general case in (Pustyl'nikov, 1999), and, in the case when \(\Pi\) is a parallelepiped in (Pustyl'nikov, 1995) in connection with the justification of the second law of thermodynamics (the law of entropy increase). From the physical point of view, GB describe a gas consisting of finitely many particles moving in a vessel, while the walls of the vessel heat up or cool down. The essence of the generalization is the following. As the particle hits the boundary \(\Gamma\ ,\) its velocity transforms with the help of a given function \(f(\gamma, t)\ ,\) defined on the direct product \(\Gamma \times \mathbb{R}^1\) (where \(\mathbb{R}^1\) is the real line, \(\gamma \in \Gamma\) is a point of the boundary and \(t \in \mathbb{R}^1\) is time), according to the following law. Suppose that the trajectory of the particle, which moves with the velocity \(v\ ,\) intersects \(\Gamma\) at the point \(\gamma \in \Gamma\) at time \(t^* \ .\) Then at time \(t^*\) the particle acquires the velocity \(v^*\ ,\) as if it underwent an elastic push from the infinitely-heavy plane \(\Gamma^*\ ,\) which is tangent to \(\Gamma\) at the point \(\gamma\ ,\) and at time \(t^*\) moves along the normal to \(\Gamma\) at \(\gamma\) with the velocity \(\frac{\partial f}{\partial t} (\gamma, t^*)\ .\) We emphasize that the position of the boundary itself is fixed, while its action upon the particle is defined through the function \(f\ .\)

We take the positive direction of motion of the plane \(\Gamma^*\) to be towards the interior of \(\Pi\ .\) Thus if the derivative \(\frac{\partial f}{\partial t} (\gamma, t) > 0\ ,\) then the particle accelerates after the impact.

If the velocity \(v^*\ ,\) acquired by the particle as the result of the above reflection law, is directed to the interior of the domain \(\Pi\ ,\) then the particle will leave the boundary and continue moving in \(\Pi\) until the next collision with \(\Gamma\ .\) If the velocity \(v^*\) is directed towards the outside of \(\Pi\ ,\) then the particle remains on \(\Gamma\) at the point \(\gamma\) until at some time \(\tilde{t} > t^*\) the interaction with the boundary will force the particle to leave it.

If the function \(f(\gamma, t)\) does not depend on time \(t\ ,\) i.e., \(\partial f/\partial t = 0\ ,\) the generalized billiard coincides with the classical one.

This generalized reflection law is very natural. First, it reflects an obvious fact that the walls of the vessel with gas are motionless. Second the action of the wall on the particle is still the classical elastic push. In the essence, we consider infinitesimally moving boundaries with given velocities.

It is considered the reflection from the boundary \(\Gamma\) both in the framework of classical mechanics (Newtonian case) and the theory of relativity (relativistic case).

Main results: in the Newtonian case the energy of particle is bounded, the Gibbs entropy is a constant (Pustyl'nikov, 1995), (Pustyl'nikov, 2005), (Deryabin and Pustyl'nikov, 2007) and in relativistic case the energy of particle, the Gibbs entropy, the entropy with respect to the phase volume grow to infinity (Pustyl'nikov, 1995), (Pustyl'nikov, 2003), references to generalized billiards.


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See also

Butterfly Effect, Chaos, Dynamical Systems, Ergodic Theory, Hamiltonian Systems, Invariant Measure, Hyperbolic Dynamics, Kolmogorov-Arnold-Moser Theory, Kolmogorov-Sinai Entropy

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