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In Section 2 we use symplectic nature of the issue and show a hyperlink of billiard ball map with geodesics on the surface. Alhazen's billiard problem seeks to search out the point at the edge of a circular "billiards" table at which a cue ball at a given level must be aimed to be able to carom once off the sting of the desk and strike another ball at a second given level. S. Thus the part house of oriented traces intersecting S?S is isomorphic to unit (co-)ball bundle of S?S. 13. The superposition of the final a hundred collisions of a thousand walkers evolved beneath LWD for 10000 collisions is proven in (crimson) darker points on prime of the part house. "is the set of periodic factors dense for a "general" easy billiard desk ? Firstly, the brand new class launched in this paper and conjectured (for an open set of parameter values) to be ergodic consists of convex planar billiards. Real billiards can contain spinning the ball so that it does not journey in a straight line, but the mathematical research of billiards generally consists of reflections wherein the reflection and incidence angles are the identical. Given a rectangular billiard table with solely corner pockets and sides of integer lengths and (with and comparatively prime), a ball despatched at a angle from a corner shall be pocketed in another nook after bounces (Steinhaus 1999, p.

Moreover, in greater dimensions billiard ball map remains to be a twist map in these symplectic coordinates and has a quite simple producing function which we shall derive below. Moreover, this geodesic curve has a principal course at every level where it passes. 2 is (up to affine similarity) one level - Th.5.3(2). One would like to calm down the situations of Hopf rigidity ultimately. 0 which implies that this geodesic has principal path all over the place on its way. The simplest method to slender down your search is to determine a price range. We additionally apply the powerful numerical method of Lyapunov-weighted dynamics to carry out a extra stringent search for elliptic islands. In general, in the rational case, one can't say extra since any two rectangles are order equivalent. A rational non-exceptional triangle order-equivalence class is (as much as the similarity) one point - Cor.5.2. One may consider billiard paths on polygonal billiard tables.

2 we can converse - as much as rotation - about horizontal, resp. 1) We denote the sides of P?P, resp. "side" e?e of P?P, resp. We name an order equivalence class a non-exceptional (resp. ?d. Let y?y, resp. Let us summarize our fundamental outcomes in terms of order-equivalence lessons. Concerning the order-equivalence of polygonal billiards, the reader can ask possible properties of corresponding equivalence classes. We examine this equivalence relation with extra regularity conditions on the orbit. The reader can verify that it's reflexive, symmetric and transitive, i.e., it's an equivalence relation. Can one replace the assumption the orbit is generic by the orbit being non-distinctive in Theorem 5.3? Size is one in every of the simplest but most necessary factors when selecting a pool desk. They encompass a degree particle which collides elastically with the walls of a bounded region, the billiard table; the shape of the desk determines the kind of dynamics which is noticed.

The case when these two segments are non-parallel (and, correspondingly, one of the arcs is shorter, with the other being longer than a semicircle) is usually referred to because the skewed stadium or squash billiard table. In arithmetic one usually desires to know if one can reconstruct a object (typically a geometric object) from sure discrete data. Can one replace the assumption that both orbits are dense (Definition 2.1 (i)) by just one orbit being dense? Even though this code shouldn't be distinctive, it is beneficial since it highlights the position of the singularities, as far as the existence and stability features of periodic orbits are involved. We proceed to research some options of the section portrait, in particular, the regions of existence and stability for the periodic orbits. 2 orbit and its corresponding island disappear, however different islands, corresponding to periodic orbits of upper durations, appear in certain areas of the (b,c)??(b,c) plane.