On the dynamics of inverse magnetic billiards

Abstract

A magnetic arc. (b) An example of two trajectories with the same value for 2 where χ and χ are supplementary.. .. .. .. 8 2.3 The behavior of the return map for fixed s 0 and varying u 0 when µ < ρ min. The Larmor centers are in orange and the dark purple points are P 0 and the corresponding P 1 , P 2 for each value of u 0. . 2.4 (a) A (2,4) periodic orbit for an ellipse with semi-major axis 3, semiminor axis 2, µ = 4/5 < ρ min = 4/3, and (s 0 , u 0) ≈ (1.3796, −0.491598);

On some refraction billiards

“…A question of great interest in the study of billiards is that of caustics, which plays a key role in the determining the regions of the plane where the orbits can access. A caustic is a smooth closed curve Γ such that every trajectory which is tangent to Γ in a point remains tangent to the latter after every passage in and out the domain D. The issue of the existence of caustics in standard billiards ( [25,33]) and its variants ( [16]) has been widely studied; in particular, in the framework of a standard convex billiard D, Lazutkin used the KAM approach to prove that, if ∂D is sufficiently smooth (of class C 553 in the original paper [25], later improved to C 6 by Douady in [14]), then there exists a discontinuous family of caustics in a small neighborhood of ∂D.…”

“…There is a wide literature on Birkhoff billiards, with recent relevant advances (see the book [33] and papers [22,23,20,4]), including some cases of composite billiard with reflections and refractions [3], also in the case of a periodic inhomogenous lattice [17]. Special mention should be paid to the work on magnetic billiards, where the trajectories of a charged particle in this setting are straight lines concatenated with circular arcs of a given Larmor radius [15,16]. Let us add that, compared with the cases quoted above, additional difficulties arise because the corresponding return map is not globally well defined and from the singularity of the Kepler potential.…”

Linear Stability of Periodic Trajectories in Inverse Magnetic Billiards

“…We give a brief review of inverse magnetic billiards in a convex set Ω ⊂ R 2 and note that additional details can be found in [10,11].…”

“…The magnetic billiard of interest is that of inverse magnetic billiards, following the naming by [22], which has only been studied in detail recently [10,11]. Given a connected domain Ω ⊂ R 2 , define a constant magnetic field B orthogonal to the plane which has strength 0 on Ω and strength B = 0 on its complement.…”

On the Dynamics of Inverse Magnetic Billiards

“…The proof that T satisfies the twist condition ∂s2 ∂u0 > 0 whenever µ < ρ min is a small exercise in geometry and trigonometry. The full proof can be found in the appendices of [6].…”