Elliptic billiard with harmonic potential: Classical description

Abstract

The classical dynamics of the isotropic two-dimensional harmonic oscillator confined by an elliptic hard wall is discussed. The interplay between the harmonic potential with circular symmetry and the boundary with elliptical symmetry does not spoil the separability in elliptic coordinates; however, it generates non-trivial energy and momentum dependencies in the billiard. We analyze the equi-momentum surfaces in the parameters space and classify the kinds of motion the particle can have in the billiard. The winding numbers and periods of the rotational and librational trajectories are analytically calculated and numerically verified. A remarkable finding is the possibility of having degenerate rotational trajectories with the same energy but different second constant of motion and different caustics and periods. The conditions to get these degenerate trajectories are analyzed. Similarly, we show that obtaining two different rotational trajectories with the same period and second constant of motion but different energy is possible.

arXiv comment15 pages, 9 figures

References in corpus (8)

• Chaotic Dirac billiard in graphene quantum dots
• Elliptic Quantum Billiard
• Periodic Chaotic Billiards: Quantum-Classical Correspondence in Energy Space
• Symmetry Breaking and Bifurcations in the Periodic Orbit Theory: I: Elliptic Billiard
• Topology of the elliptical billiard with the Hooke's potential
• Periodic ellipsoidal billiard trajectories and extremal polynomials
• Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?
• Integrable elliptic billiards and ballyards

Cited by in corpus (2)

• Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries
• Quantum states resembling classical periodic trajectories in mesoscopic elliptic billiards