Abstract
We discuss Liouville's theorem for nonsmooth integrable systems of the billiard type and give a scheme of calculation of angle-action variables for the flow. We also deal with the problem of pseudointegrability. We discuss the relationship between the continuous-time (flow) and the discrete-time (map) approaches. We treat all these aspects through a specific billiard—a wedge embedded in a two-dimensional isotropic harmonic potential. Varying the parameters provides two integrable and two pseudointegrable cases. It turns out that the dynamics of one of the latter is indeed integrable in a certain sense. We also address the problem of applying perturbation theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. V. Berry, Regularity and chaos in classical mechanics, illustrated by three deformations of a circular “billiard,”Eur. J. Phys. 2:91 (1981).
P. J. Korsch and J. Lang, A new integrable gravitational billiard,J. Phys. A 24:45 (1991).
J. P. Eccmann and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys. 57:617 (1985).
L. A. Bunimovich, Decay of correlations in dynamical systems with chaotic behavior,Sov. Phys. JETP 62:842 (1985).
P. J. Richens and M. V. Berry, Pseudointegrable systems in classical and quantum mechanics,Physica 2D:495 (1990).
V. I. Arnold,Mathematical Methods of Classical Mechanics, 2nd ed., (Springer-Verlag, Berlin, 1989).
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,Ergodic Theory (Springer-Verlag, Berlin, 1982).
J. D. Meiss, Symplectic maps, variational principles, and transport,Rev. Mod. Phys. 64(3):795 (1992).
T. Dagaeff and C. Rouvinez, On the discontinuities of the boundary in billiards,Physica D 67:166 (1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dagaeff, T. Integrability and pseudointegrability in billiards illustrated by the harmonic wedge. J Stat Phys 83, 39–70 (1996). https://doi.org/10.1007/BF02183639
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02183639