A special billiard trajectory inside
a triaxial ellipsoid Q: it
consists of segments of generatrices of a one-napple hyperboloid H
confocal to Q. The trajectory is parameterized in
terms of sigma-functions of one variable and exponents.
(Figure created by CorelDraw10).
(Figure created by CorelDraw10).
A billiard with a Hooke
potential inside an ellipse Q:
the trajectories between the bounces are arcs of conics. Before and after impacts,the conics are tangent to the same ellipses Q1, Q2 confocal to Q.
(The figure created by CorelDraw 10).
The original reference is
the trajectories between the bounces are arcs of conics. Before and after impacts,the conics are tangent to the same ellipses Q1, Q2 confocal to Q.
(The figure created by CorelDraw 10).
The original reference is
A detailed derivation of the solutions is in
PDF file here .
Fully bi-asimptotic solutions to the
billiard problem can be found in
Another intersting
integrable generalization of the classical ellipsoidal billiard is the
billiard on a quadric. Some related references are
Chang, Shau-Jin; Shi, Kang
Jie. Billiard systems on quadric surfaces and the Poncelet theorem.
J. Math. Phys.
30 (1989), no. 4, 798--804.
Dragovic, Vladimir;
Radnovic, Milena. Geometry of integrable billiards and pencils of
quadrics. J. Math. Pures Appl.
(9) 85 (2006), no. 6, 758--790
Ellipsoidal
Billiards
The
straight line billiard
inside a quadric in Rn
is a very well studied classical discrete integrable system. Everything
is known: equations of the billiard map, their Lax representations, the
general solution in terms of theta-function, and asymptotic
solutions. Yet, there are several interesting details and
generalizations worth to mention.