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Figure 1: Sinai billiards
Dynamical Billiard is a
corresponding to the inertial motion of a point
mass within a region \(\Omega\) that has a
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).
The orbits of billiards are broken lines in
\(\Omega\) 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
then the orbits consist
of geodesic (rather than straight) segments. Billiard is a
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
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
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
from integrable to completely
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.
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.
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)
(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
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.
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 , 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).
Most classes of billiards demonstrate
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 .
Therefore a key question in the theory of billiards is concerned
with the mechanisms of chaos in these systems.
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
statistical properties of
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 ).
Sinai billiards have the strongest possible chaotic properties, being ,
, , having a positive
and an exponential . 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
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 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 ).
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.
Figure 3: Mushroom
It is a general belief that
the phase spaces of typical Hamiltonian systems are divided into -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 ).
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,
Many properties of classical dynamics of billiards are closely related to the
properties of the corresponding quantum problem. Consider the stationary
\(H\psi=E\psi\) 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).
Berger M. (1995) Seules les quadriques admettent des caustiques. Bull. Soc. Math. France 123:107-116.
Birkhoff G. (1927) Dynamical Systems. American Mathematical Society Colloquium Publication, 9.
Boldrighini C., Keane M. & Marchetti F. (1978) Billiards in polygons. Annals of Probability, 6:532-540.
Bunimovich L. A. (1974a) On billiards close to dispersing. Mathematical USSR Sbornik, 95:49-73 (originally published in Russian).
Bunimovich L. A. (1974b) The ergodic properties of certain billiards. Funkt. Anal. Prilozh. 8:73-74.
Bunimovich L. A. (1992) On absolutely focusing mirrors. Springer Lecture Notes in Mathematics, .
Bunimovich L. A. (2000) Billiards and other hyperbolic systems with singularities. In Dynamical Systems, Ergodic Theory and Applications, edited by Ya. G. Sinai, Berlin: Springer.
Bunimovich L. A. (2001) Mushrooms and other billiards with divided phase space. Chaos, 11:802-808.
Bunimovich L. A. & Rehacek J. (1998) How high-dimensional stadia look like. Communications in Mathematical Physics, 197:277-301.
Burago D., Ferleger S. and Kononenko A. (2000) A geometric approach to semi-dispersing billiards. In Hard Ball Systems and the Lorentz Gas, edited by D. Szasz, Berlin: Springer.
Chernov N. I. & Markarian R. (2006) Chaotic Billiards, American Mathematical Society, Mathematical Surveys and Monographs, vol. 127.
Donnay V. (1991) Using integrability to produce chaos: billiards with positive entropy. Communications in Mathematical Physics, 141:225-257.
Farber M., Tabachnikov S. (2002) Topology of cyclic configuration spaces and periodic orbits of multi-dimensional billiards, Topology, 41: 553-589.
Kerckhoff S., Mazur H. & Smillie J. 1986. Ergodicity of billiard flows and quadratic differentials. Annals of Mathematics, 124:293-311.
Kozlov V. V. and Treshchev D. V. (1991) Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts, American Mathematical Society, Translations of Mathematical Monographs, vol. 89.
Lazutkin, V. F. (1973) The existence of caustics for a billiard problem in a convex domain. Math. USSR Izvestija 7:185-214.
Lazutkin V. F. (1991) The
and Asymptotics of Spectrum of Elliptic Operators. Berlin and New York: Springer.
Markarian R. (1988) Billiards with Pesin region of measure one. Communications in Mathematical Physics 118:87-97.
Masur H., Tabachnikov S. (2002) Rational billiards and flat structures. In Handbook of Dynamical Systems, vol. 1A, edited by B. Hasselblatt and A. Katok, Amsterdam: Elsevier.
Mather J. (1984) Non-existence of invariant circles. Ergod. Th. Dyn. Syst. 4:301-309.
Shnirelman A. I. (1991) On the asymptotic properties of eigenfunctions in the regions of chaotic motion.
Addendum in The KAM Theory and Asymptotics of Spectrum of Elliptic Operators by V. F. Lazutkin.
Berlin and New York: Springer.
Simanyi, N. (2003) Proof of the Boltzamnn-Sinai ergodic hypothesis for typical hard disk systems. Inventiones Math., 154:123-178.
Sinai Ya. G. (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Mathematical Surveys, 25:137-189 (originally published in Russian 1970).
Tabachnikov S. (1995) Billiards, Societe Mathematique de France. "Panoramas et Syntheses," No. 1.
Tabachnikov S. (2005) Geometry and Billiards, American Mathematical Society Press.
Wojtkowski M. (1986) Principles for the design of billiards with nonvanishing . Communications in Mathematical Physics, 105:391-414.
Internal references
James Meiss (2007) . , 2(2):1629.
Tomasz Downarowicz (2007) . Scholarpedia, 2(11):3901.
James Meiss (2007) . Scholarpedia, 2(8):1943.
Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) . Scholarpedia, 1(7):1358.
Philip Holmes and Eric T. Shea-Brown (2006) . Scholarpedia, 1(10):1838.
Animation of dispersing billiard
Article #58. animation (6.1M) (Carl Dettman)
Animation of the stadium billiard
(David Harrison)
Simulation of mushrooms and other billiards
(Steven Lansel)
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