Hard Chaos in Magnetic Billiards¶(On the Euclidean Plane)

Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

“…Magnetic billiards on the plane have been considered in many works [BR], [K], [BK], [Ta1] and on the hyperbolic plane in [Ta2]. The study of such billiards is strongly motivated by mesoscopic physics, where such billiard models are used as simplified version of the mesoscopic devices in the presence of magnetic fields.…”

“…Let m be the bouncing point, let C be the osculating circle at m and let m ′ be the second point in which the particle trajectory intersects C. Denote by χ − , χ + the curvatures of the infinitesimal beam immediately before and after reflection. Then, the change of the curvature under the reflection [Ta1], [K] at the bouncing point m (see fig. 1b) is given by…”

“…Using classical formulas for surfaces of constant curvature ( [Vi], see also [Ta1], [Ta2]), it is possible to give a geometric interpretation of the function K(·). Let v ∈ V , and let l ∈ ∂Q be the origin point of v. Let C(l) ⊂ M be the osculating circle (resp.…”

Hard chaos in magnetic billiards (on the hyperbolic plane)

“…10,5,11 The presence of magnetic field changes drastically the dynamics of the billiard system; it breaks time reversal symmetry, diminishes or completely suppresses hyperbolicity in the bounceless segments, and strongly influences the effects of the bounces. [12][13][14] Previously there has been work done on classical [15][16][17] and quantum billiards 18 with a magnetic field, even on curved spaces. 19 Recently the study of planar magnetic billiards gains considerable inspiration from mesoscopic physics, since under certain conditions the motion of electrons in two dimensional mesoscopic systems can be approximated with high accuracy by classical or semiclassical methods, and the transport properties of these systems can also be investigated experimentally in the presence of a magnetic field as well.…”

“…13,15,16,17 in the Euclidean case. In general this linear mapping depends on several parameters ͑the flying time, the magnetic field, the angle of incidence, and the curvature of the billiard wall at the bouncing point͒, nevertheless, a simple geometric rule can be given for determining the stability character of the map in hand.…”