Jump to Content
Your Recent History
View Cart
Logged Out Log In
SIAM J. Appl. Dyn. Syst. 10, pp. 278-324 (47 pages)
The Frequency Map for Billiards inside Ellipsoids
Alerts
Alert Me When Cited
Alert Me When Corrected
Tools
Download Citation
Add to MyScitation
Blog This Article
Print-Friendly
Research Toolkit
Share
Email Abstract
Connotea
CiteULike
del.icio.us
BibSonomy
Tweet this Article
Add to FacebookThe billiard motion inside an ellipsoid $Q\subset\mathbb{R}^{n+1}$ is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each Liouville torus becomes just a parallel translation with some frequency $\omega$ that varies with the torus. Further, any billiard trajectory inside $Q$ is tangent to $n$ caustics $Q_{\lambda_1},\ldots,Q_{\lambda_n}$, so the caustic parameters $\lambda=(\lambda_1,\ldots,\lambda_n)$ are integrals of the billiard map. The frequency map $\lambda\mapsto\omega$ is a key tool for understanding the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present two conjectures, fully supported by numerical experiments. We obtain, from one of the conjectures, some lower bounds on the periods. These bounds depend only on the type of the $n$ caustics. We describe the geometric meaning, domain, and range of $\omega$. The map $\omega$ can be continuously extended to singular values of the caustic parameters, although it becomes “exponentially sharp” at some of them. Finally, we study triaxial ellipsoids of $\mathbb{R}^3$. We numerically compute the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of $\mathbb{R}^2$.
© 2011 Society for Industrial and Applied Mathematics
RELATED DATABASES
To view database links for this article,
you need to log in.
KEYWORDS
ARTICLE DATA
History
Received April 30, 2010
Accepted January 18, 2011
Published online March 24, 2011
Accepted January 18, 2011
Published online March 24, 2011
Digital Object Identifier
For access to fully linked references, you need to log in.
ALL SIAM Content
Scitation
Google Scholar