Numerical study of a billiard in a gravitational field |
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| Institution: | 1. Resident, Department of Stomatology, The Second Xiangya Hospital, Central South University, Changsha, Hunan, China;2. Resident, Department of Stomatology, The Second Xiangya Hospital, Central South University, Changsha, Hunan, China;3. Visiting Physician, Department of Stomatology, The Second Xiangya Hospital, Central South University, Changsha, Hunan, China;4. Resident, Department of Stomatology, The Second Xiangya Hospital, Central South University, Changsha, Hunan, China;6. Professor, Department of Stomatology, The Second Xiangya Hospital, Central South University, Changsha, Hunan, China;5. Associate Professor, Department of Stomatology, The Second Xiangya Hospital, Central South University, Changsha, Hunan, China;1. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de Mexico, D.F. 04510 México, Mexico;2. Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan;3. Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA;4. TRIUMF, Vancouver, BC, Canada V6T 2A3;5. Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1;6. University of Northern British Columbia, Prince George, BC, Canada V2N 4Z9;7. KEK, Tsukuba-shi, Ibaragi, Japan;8. Brookhaven National Laboratory, Upton, NY 11973-5000, USA;9. Arizona State University, Tempe, AZ 85287-1504, USA |
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| Abstract: | Billiards have always been used as models for mechanical systems. In this paper we describe a very simple billiard which, over a range of one continous parameter only, exhibits the characteristics of Hamiltonian systems having two degrees of freedom and a discontinuity. The relationship between this billiard and the well-known one-dimensional self-gravitating system (with N = 3) is given. This billiard consists of a mass point moving in a symmetric wedge of angle 2θ under the influence of a constant gravitational field. For θ<45° KAM and chaotic regions coexist in the phase space. A specific family of curves, related to collisions at the wedge vertex, limits the expansion of near-integrable regions. For θ=45°, the motion is strictly integrable. Finally, for θ>;45°, complete chaos is obtained, suggesting K-system behavior. The general properties of the mapping and some numerical results obtained are discussed. Of special interest are invariant curves which cross a line of discontinuity, and a new “universality” class for Lyapunov numbers. |
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