On some refraction billiards

Abstract

<p style='text-indent:20px;'>The aim of this work is to continue the analysis, started in [<xref ref-type="bibr" rid="b10">10</xref>], of the dynamics of a point-mass particle <inline-formula><tex-math id="M1">\begin{document}$ P $\end{document}</tex-math></inline-formula> moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> is divided into two complementary domains, depending on whether the gravitational effects of the galaxy's mass distribution or of the Black Hole prevail. Thus, solutions alternate arcs of Keplerian hyperbolæ with harmonic ellipses; at the interface, the trajectory is refracted according to Snell's law. The model was introduced in [<xref ref-type="bibr" rid="b11">11</xref>], in view of applications to astrodynamics.</p><p style='text-indent:20px;'>In this paper we address the general issue of periodic and quasi-periodic orbits and associated caustics when the domain is a perturbation of the circle, taking advantage of KAM and Aubry-Mather theories.</p>

Chaotic dynamics in refraction galactic billiards

“…Our results take advantage of the fact that a Keplerian centre acts as a scatterer at high energies (see e.g. [8,10]) and complement the almost-integrability of the model proved in [13].…”

“…In this work, along with the previous papers [12,13], we presented the analysis of a brand new dynamical model of interest in Celestial Mechanics, starting from the basic study of its fixed points and arriving to its non-integrability. In particular there is fil rouge between the first and the present paper: an elliptic domain with its center in the origin satisfies the assumptions of theorem 4.7 and thus the associated system is chaotic; this represents the analytical proof of the numerical results shown in paper [12, figure 11].…”

“…Their physical interest cover many different situations (see for instance [19,30]). In particular, the dynamical system considered in the present paper is of interest in Celestial Mechanics and involves two forces acting in two complementary regions of the plane: a Keplerian center of gravitation sits inside a bounded region D, while a harmonic oscillator is acting in the complementary set of D. This choice of the potentials appears in the literature (see [12][13][14]) to mimic the motion of a particle in an elliptic galaxy with a Black Hole in its centre; in particular in [14], the 3-dimensional model is considered and its chaoticity is inferred using a mixed numerical and analytical approach. At the interface of the two regions, which we assume to be a smooth curve, a generalized Snell's refraction law holds and trajectories concatenate inner arcs (Keplerian hyperbolae) with outer ones (harmonic arcs) being deflected in the transition through the boundary of D. This indeed corresponds to some (possibly ill-defined) area-preserving map in the cylinder.…”

Stack-sorting for Coxeter groups

“…we obtain a Markov chain M (α,β,t) e 1 on C n × Z/2nZ, which models a certain random combinatorial billiard trajectory in the n-dimensional torus T = V * /Q ∨ = R n /Z n (see Figure 17). This Markov chain is isomorphic to ▲ASEP ob λ ; the isomorphism is just the map C n × Z/2nZ → C n × ±[n] given by (w, i) → (w, ι −1 (i)), where ι : ± [n] → Z/2nZ is the bijection defined in (35). Hence, it follows from Theorem 7.4 that the stationary probability of a state (w, i) in M G w (χ (ι −1 (i)) ; t) K λ (χ; t) .…”