Projective Integrable Mechanical Billiards

Abstract

In this paper, we use the projective dynamical approach to integrable mechanical billiards as in [22] to establish the integrability of natural mechanical billiards with the Lagrange problem, which is the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the middle of the Kepler centers, as the underlying mechanical systems, and with any combinations of confocal conic sections with foci at the Kepler centers as the reflection wall, in the plane, on the sphere, and in the hyperbolic plane. This covers many previously known integrable mechanical billiards, especially the integrable Hooke, Kepler and two-center billiards in the plane, as has been investigated in [19], as subcases. The approach of [19] based on conformal correspondence has been also applied to integrable Kepler billiards in the hyperbolic plane to illustrate their equivalence with the corresponding integrable Hooke billiards on the hemisphere and in the hyperbolic plane as well.

Chaotic dynamics in refraction galactic billiards

“…Thus, also in this case, the dynamics associated to a centered ellipse is chaotic. This negatively complements the recent results about integrability of the focused elliptic Kepler billiard by Takeuchi and Zhao [36][37][38], namely, with the singularity in one of the two foci. This is coherent with our study, since focused ellipses, having only two central configurations which are antipodal, are not admissible domain.…”

“…Taking again the example of an elliptic domain, we deduce that, as far as the singularity is in the center of the ellipse, the reflective system is chaotic as well. This negatively complements the recent results by Takeuchi and Zhao [36][37][38] where they consider an elliptic Kepler billiard with the mass in one of the foci, proving its integrability. Note that an ellipse with focus at the origin does not satisfy the hypotheses of theorem 1.4, while it does when moving the gravitational center at the center of the ellipse.…”