Billiard in a rotating half-plane
The main objective of this research is to study the properties of a billiard system in an unbounded domain with moving boundary. We consider a system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points and exp...
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| Veröffentlicht in: | Journal of dynamical and control systems 2023-10, Vol.29 (4), p.1695-1707 |
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| container_title | Journal of dynamical and control systems |
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| creator | Kryzhevich, Sergey Plakhov, Alexander |
| description | The main objective of this research is to study the properties of a billiard system in an unbounded domain with moving boundary. We consider a system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points and experiences elastic collisions with the ball. We define a mathematical model for the dynamics of such a system and write down asymptotic formulae for its motions. In particular, we determine existence and uniqueness of solutions. We find all possible grazing impacts of the ball. Besides, we demonstrate that for almost every initial condition, the ball goes to infinity exponentially fast, with the time intervals between neighboring collisions tending to zero. The approach developed in this paper is an original combination of methods of Billiards and Vibro-Impact Dynamics. It could be a base for studying more complicated systems of similar types. |
| doi_str_mv | 10.1007/s10883-023-09655-z |
| format | Article |
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| source | SpringerNature Journals |
| subjects | Calculus of Variations and Optimal Control Optimization Collisions Control Dynamical Systems Dynamical Systems and Ergodic Theory Elastic scattering Half planes Mathematics Mathematics and Statistics Original Research Straight lines Systems Theory Vibration |
| title | Billiard in a rotating half-plane |
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