Conformal Transformations and Integrable Mechanical Billiards

Projective Integrable Mechanical Billiards

“…In this paper, we explain that certain integrable mechanical billiards in the two-dimensional plane and constant curvature surfaces are related by projective correspondences. This allows us to yet extend some of our previous results in [19] concerning integrable mechanical billiards in the plane with further extensions to surfaces of constant curvatures.…”

“…The integrability of such systems has been established recently by Gallavotti and Jauslin [7] with an analysis on the geometry of ellipses, with alternative proofs by [6] and [22]. In [19], we establish that any conic sections focused at the center and any confocal combination of them are also integrable, by using the classical Hooke-Kepler correspondence.…”

“…As compared to the Hooke and the Kepler problems, Euler's two-center problem in R 2 are not super-integrable and the billiard problem defined by it seems to be less studied. In [19], we showed the integrability of such billiards with combinations of confocal conic sections reflection walls.…”

“…In the plane, the billiard problems defined through the Hooke, the Kepler, and the two-center problems with combinations of confocal conic sections are therefore subcases of Theorem 1 and thus their integrability directly follows. These have been previously discussed via a different method, based on conformal transformations, in [19]. Theorem 1 provides an alternative proof of their integrability as well as extensions to the sphere and the hyperbolic plane.…”

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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.

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