Presentation
Fitting Invariant Curves on Billiard Tables and the Birkhoff-Herman Theorem
IV International Symposium on Hamiltonian Systems and Celestial Mechanics, HAMSYS-2001
(2004)
Abstract
A two-dimensional billiard table is geometrically integrable when the phase space is foliated by continuous invariant curves. When an integrable planar domain has a C⁴ boundary with strictly positive curvature, a neighborhood of the boundary is foliated by invariant circles. This family of invariant circles can lose convexity only after developing a singularity and if it developes a singularity, the boundary contains a segment of an ellipse. An important role in this result is played by the Birkhoff-Herman thoerem which shows that differentiability of enveloped curves cannot be lost without a change in homotopy type.
Keywords
- Billiard map,
- Integrable,
- Invariant curves,
- Birkhoff-Herman theormen
Disciplines
Publication Date
January 1, 2004
DOI
10.1007/978-1-4419-9058-7_2
Comments
NewAdvances in Celestial Meclumics and Hamiltonian Systems, edited by Delgado et al., Kluwer Academic/Plenum Publishers, New York, 2004
Citation Information
Edoh Y. Amiran. "Fitting Invariant Curves on Billiard Tables and the Birkhoff-Herman Theorem" IV International Symposium on Hamiltonian Systems and Celestial Mechanics, HAMSYS-2001 (2004) Available at: http://works.bepress.com/edoh_amiran/7/