# The existence of caustics for a billiard problem in a convex domain

TL;DR: In this paper, a system of caustics is found for a plane convex domain with a sufficiently smooth boundary; the Caustics are close to the boundary and occupy a set of positive measure.

Abstract: A system of caustics is found for a plane convex domain with a sufficiently smooth boundary; the caustics are close to the boundary and occupy a set of positive measure. A caustic is a convex smooth curve lying in the domain and possessing the property that a tangent to it becomes another tangent to the same curve after reflection from the boundary according to the law of geometrical optics.

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TL;DR: The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state associated with stochastic classical motion in all or part of the (2N-1) energy surface in phase space as discussed by the authors.

Abstract: The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.

1,068 citations

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TL;DR: In this paper, the B-property for two-dimensional domains with focusing and neutral regular components is proved and some examples of three and more dimensional domains with billiards obeying this property are also considered.

Abstract: For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditionsB-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.

574 citations

### Cites background from "The existence of caustics for a bil..."

...However in the paper of Lazutkin [7] it was shown that a billiard inside a sufficiently smooth (C, /c^ 553) convex curve in the two - dimensional plane possess infinitely many caustics which fill in a set of positive measure....

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...known that the billiard system inside the domain bounded by a smooth plane convex curve has a set of positive measure consisting of the tori [6, 7]....

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01 Jan 2005TL;DR: In this article, the authors discuss the existence and non-existence of caustics and periodic trajectories of billiards inside conics and quadrics, as well as in polygons.

Abstract: Motivation: Mechanics and optics Billiard in the circle and the square Billiard ball map and integral geometry Billiards inside conics and quadrics Existence and non-existence of caustics Periodic trajectories Billiards in polygons Chaotic billiards Dual billiards Bibliography Index.

403 citations

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01 Jan 2002

TL;DR: In this paper, the authors describe elementary celestial and Hamiltonian mechanics and a quasi-integrable Hamiltonian system for the detection of chaos in the universe. But they do not describe the physical structure of the world.

Abstract: 1. Elementary Celestial and Hamiltonian Mechanics 2. Quasi-Integrable Hamiltonian System 3. Kam Tori 4. Single Resonance Dyanmics 5. Numerical Tools for the Detection of Chaos 6. Interactions Among Resonances 7. Secular Dynamics of the Planets 8. Secular Dynamics of Small Bodies 9. Mean Motion Resonances 10. Three Body Resonances 11. Secular Dynamics Inside Mean Motion Resonances 12. Global Dynamical Structure of the Belts of Small Bodies.

401 citations

### Cites background from "The existence of caustics for a bil..."

...A theoretical result by Lazutkin (1973) shows that the frequencies on KAM tori can be fitted by a smooth (C∞) function of the actions....

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