Tangent map for classical billiards in magnetic fields

Meplan, O; Brut, F.; Gignoux, C.

doi:10.1088/0305-4470/26/2/012

Cited by 24 publications

(21 citation statements)

References 7 publications

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“…Simplifying the problem further, in this section we deal with a single product of two matrices of the form R(A)L(t), which is sometimes called the tangent map or the Jacobian of the bouncing map. [15][16][17] It can be considered as the stability matrix of special trajectories having one bounce translational symmetry ͑examples are given in the next section͒, or it can also be regarded as the building stone of longer matrix products.…”

Section: A Geometrical Theorem On Local Stabilitymentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Hard chaos in magnetic billiards (on the hyperbolic plane)

Tasnádi

1998

Journal of Mathematical Physics

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Dynamics of a gravitational billiard with a hyperbolic lower boundary Chaos 9, 841 (1999);In this paper some results on the local and global stability analysis of magnetic billiard systems, established on two dimensional Riemannian manifolds of constant curvature are presented, with particular emphasis on the hyperbolic plane. For special billiards, possessing a discrete group of ͑rotational or translational͒ symmetry, a geometrical theorem, illustrated by numerical simulations, is given on the stability of trajectories with the same symmetry. We also present sufficient criteria for the global hyperbolicity of the dynamics ͑hard chaos͒, and give lower estimations for the Lyapunov exponent in terms of the shape of the billiard.

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Section: A Geometrical Theorem On Local Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hard chaos in magnetic billiards (on the hyperbolic plane)

Tasnádi

1998

Journal of Mathematical Physics

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“…Billiard in constant magnetic and electric fields .-The billiards in magnetic field were first considered in [18] and later studied in [7,8,19]. We use our result to provide a criterion of stability of the solutions near the boundary.…”

Section: Vadim Zharnitskymentioning

confidence: 98%

Quasiperiodic Motion in the Hamiltonian Systems of the Billiard Type

Zharnitsky

1998

Phys. Rev. Lett.

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It is shown that two-degree-of-freedom Hamiltonian systems of the billiard type are equivalent to adiabatically varying one-degree-of-freedom Hamiltonian systems for solutions staying near the boundary. Under some nondegeneracy conditions such systems possess a large set of quasiperiodic solutions filling out two-dimensional invariant tori. The latter separate the extended phase space into layers providing stability for all time. The result is illustrated on a few examples. [S0031-9007(98)07815-6] PACS numbers: 05.45. + b, 02.30.Hq, 03.20. + i Billiards have been frequently used to investigate the relation between the classical and quantum mechanics. This relation is clear for exactly solvable systems but is less understood if the classical system has both regular and chaotic components in the phase space. A nonelliptic convex billiard provides one of the simplest examples of such systems. Indeed, since in billiards the particle interacts only with a one-dimensional boundary and is free in the interior, it makes analysis and numerical simulations easier. On the other hand, a convex billiard is known to possess a large family of caustics (defined as the envelopes of light ray trajectories) accumulating at the boundary [1]. The caustics represent invariant sets in the phase space and, thus, are important for constructing the quasimodes and estimating the corresponding eigenvalues for the quantum billiard [2,3].Such near integrable behavior in the vicinity of the boundary of a convex billiard can be anticipated by noting that a trajectory nearly tangent to the boundary will experience many collisions with the boundary before the curvature will significantly change. This raises a hope of introducing different time scales and obtaining an adiabatic invariant.An adiabatic invariant, which is given by the action variable, is well known to exist in slowly varying smooth Hamiltonian systems H͑q, p, l͒, where l et [4,5]. Moreover, it was shown by Arnold [6] that in a nonlinear system with a slowly periodically varying smooth Hamiltonian the adiabatic invariant is conserved perpetually under some nondegeneracy conditions. The proof is based on the application of Kolmgorov-Arnold-Moser (KAM) theory to a reduced Hamiltonian obtained from the original Hamiltonian H͑q, p, l͒ after a series of canonical transformations are carried out. This result showed that the majority of slowly periodically varying smooth Hamiltonian systems possess a large set of invariant tori carrying quasiperiodic motion.Contrary to the smooth case, the stability problem for the systems with impacts had to be resolved on a case by case basis, see [2,[7][8][9][10][11], by first obtaining a "bouncing map" and then bringing it to the near integrable form.In this Letter, we use the adiabatic invariant approach to establish a counterpart of Arnold's result for twodimensional billiard systems (or equivalently for slowly periodically varying one-dimensional impact oscillators) providing a unified approach to investigate the regularity of motion in the system...

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“…The main distinction here from other works on annular billiards is that in the rotating frame the electron moves along curved paths between successive wall collisions. The introduction of a Coriolis term into the Hamiltonian upon the transformation to the rotating frame is, of course, akin to introducing an effective uniform magnetic field (and another frequency, the Larmor frequency) in which an electron moves on a curved path [15][16][17][18][19]. The laser wavelength is taken as 780 nm (corresponding to a frequency of 0.0584 a.u.)…”

Section: Introductionmentioning

confidence: 99%

Annular billiard dynamics in a circularly polarized strong laser field

et al. 2012

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We analyze the dynamics of a valence electron of the buckminsterfullerene molecule (C60) subjected to a circularly polarized laser field by modeling it with the motion of a classical particle in an annular billiard. We show that the phase space of the billiard model gives rise to three distinct trajectories: "whispering gallery orbits," which hit only the outer billiard wall; "daisy orbits," which hit both billiard walls (while rotating solely clockwise or counterclockwise for all time); and orbits that only visit the downfield part of the billiard, as measured relative to the laser term. These trajectories, in general, maintain their distinct features, even as the intensity is increased from 10(10) to 10(14) Wcm-2. We attribute this robust separation of phase space to the existence of twistless tori.

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Tangent map for classical billiards in magnetic fields

Cited by 24 publications

References 7 publications

Hard chaos in magnetic billiards (on the hyperbolic plane)

Hard chaos in magnetic billiards (on the hyperbolic plane)

Quasiperiodic Motion in the Hamiltonian Systems of the Billiard Type

Annular billiard dynamics in a circularly polarized strong laser field

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