Gravitational billiards bouncing inside general domains - foci curves and confined domains
Abstract
A massive particle under the influence of a constant gravitational force that is bouncing inside an ideal reflecting mirror described by some function f (x) is considered. For the associated flight trajectories we derive the parametric curves, named foci curves. All foci points of the parabolas for a given initial position and energy lie on these curves. From these foci curves the associated flight parabola envelopes are derived resulting, together with the mirror surface, in a confined domain for all possible particle trajectories in the non-periodic orbit case. The general results are briefly discussed and visualized for three concrete mirror surfaces.
- Publication:
-
Journal of Geometry and Physics
- Pub Date:
- December 2023
- DOI:
- 10.1016/j.geomphys.2023.104998
- arXiv:
- arXiv:2211.03788
- Bibcode:
- 2023JGP...19404998J
- Keywords:
-
- Billiards;
- Gravity;
- Foci;
- Reflecting mirror;
- Flight parabola;
- Confined trajectories;
- Mathematics - Dynamical Systems;
- Mathematical Physics;
- Nonlinear Sciences - Chaotic Dynamics;
- Physics - Classical Physics;
- Physics - Optics;
- 37C83
- E-Print:
- 14 pages, 9 figures