Projective dynamics and an integrable Boltzmann billiard model

Abstract

The aim of this note is to explain the integrability of an integrable Boltzmann billiard model, previously established by Gallavotti and Jauslin [G. Gallavotti and I. Jauslin, A theorem on Ellipses, an integrable system and a theorem of Boltzmann, preprint (2020); arXiv:2008.01955], alternatively via the viewpoint of projective dynamics. We show that the energy of a corresponding spherical problem leads to an additional first integral of the system equivalent to Gallavotti–Jauslin’s first integral. The approach also leads to a family of integrable billiard models in the plane and on the sphere defined through the planar and spherical Kepler–Coulomb problems.

Projective Integrable Mechanical Billiards

“…A special property of the Kepler problem is that this remains true when Z is not vertical, up to a change of metric and of the mass factor [8]. See also [1], [22].…”

“…To show their independence, we give their explicit expressions in a common chart as in [22]. To normalize our situation, we here again assume that Z mid = (0, 0, −1), Z1 = (0, a, −1), and Z2 = (0, −a, −1).…”

“…For the Kepler problem in R 2 , the billiard systems with a line reflection wall which is not passing the center were proposed by Boltzmann [5]. 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.…”

“…The energies of the systems then give rise to a pair of independent first integrals for both of the two billiard systems. With this method, the projective correspondence between integrable planar and spherical Kepler billiards with a line or centered circle reflection wall was presented in [22]. The method can be thought of as an adaptation of the projective method for geodesic flows and free billiards as in [20], [15], [16], [17], [18] to the case of mechanical billiards.…”

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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.

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