Number of possible directions of their orbits. Recently obtained (see the review paper Masur, Tabachnikov, 2002).īilliards in rational polygons are nonergodic because of a finite A lot of impressive resultsĪbout a fine structure of billiard orbits in rational polygons were
In particular, on almost all invariant manifolds the system is uniquelyĮrgodic, i.e., has an unique invariant measure. This flow has a conserved quantity whichĪllows the application of the powerful techniques from Teichmuller theory. Therefore, one can constructĪ Riemann surface from a finite number of copies of \(\Omega\) andĬonsider there a directional flow. There exist only a finite number of copies of \(\Omega\) obtainedīy the reflections with respect to its sides. \) Each billiard orbit inĪ rational polygon can have only a finite number of directions and also Its angles are commensurable with \(\pi\. Lines by consecutive reflections of the billiard table with respect to itsįaces (sides) at each reflection. Polyhedra all smooth components of the boundary \(\partial\Omega\)Īre flat and therefore one can turn the billiard orbits into straight Of a billiard in \(\Omega\) which are separated by a reflectionįrom \(\Gamma\) corresponds a single link of the correspondingīilliard orbit in \(\Omega^*\. "straightened" in the sense that to each two consecutive links of an orbit The orbits of the billiard in \(\Omega^*\) are \(\Omega^*\) which is the union of \(\Omega\) and its Reflect \(\Omega\) with respect to \(\Gamma\) and consider a "double" billiard table Have a flat component \(\Gamma\) of the boundary \(\partial\Omega\. Inward, outward the billiard region, or if it is flat (has zero curvature) A smoothĬomponent of the boundary is dispersing, focusing or neutral if it is convex Hamiltonian systems from integrable to completely chaotic ones. The dynamics of billiards is completely defined by the shape of itsīoundary and it demonstrates all the variety of possible behaviors of
#SPACE BILLIARDS FULL#
Particular, the set of all orbits which hit singular points of the boundary of aīilliard table has phase volume zero, and therefore billiard dynamics is wellĭefined on a subset of the phase space which has a full phase volume. \) Hence, the phase volume is preserved under the dynamics and in manyĬases one can neglect such sets of orbits which have phase volume zero. Billiards models are Hamiltonian systems with potential \(V(q)\) that is equal to zero within a billiard table \(\Omega\) and infinity Point of the next reflection from the boundary. Of the point of reflection into the coordinates and the incident angle at the However, dynamics of billiards can be rather completelyĬharacterized by a billiard map which transforms coordinates and incident angle Billiard is a dynamical system withĬontinuous time. Of geodesic (rather than straight) segments. If \(\Omega\) is a region on a Riemannian manifold then the orbits consist The region \(\Omega\) is also called aīilliard table.
#SPACE BILLIARDS FREE#
Segments corresponding to the free paths within the region and the verticesĬorresponding to the reflections off its boundary. The orbits of billiards are broken lines in configuration space \(\Omega\) with the
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