Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

Gutkin, Boris

doi:10.1007/s002200000346

Cited by 24 publications

(32 citation statements)

References 7 publications

(15 reference statements)

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“…Moreover, under this assumption, if a circle of radius r oriented in the same direction as the boundary is tangent to ∂Ω (with the agreed orientation), then it contains the domain Ω inside. Ordinary Magnetic billiards were studied in many papers; see, e.g., [2], [5], [18], [3], [26], [10]. Two-sided magnetic billiards were introduced in [23].…”

Section: Magnetic Billiardsmentioning

confidence: 99%

A survey on polynomial in momenta integrals for billiard problems

Bialy

Mironov

2018

Phil. Trans. R. Soc. A.

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In this paper, we give a short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique, we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

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Section: Magnetic Billiardsmentioning

confidence: 99%

A survey on polynomial in momenta integrals for billiard problems

Bialy

Mironov

2018

Phil. Trans. R. Soc. A.

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“…It follows immediately that the equation of motion for the velocity v depends only on the rotation B = ∇×A of the vector potential. It reads m er = qB ∇(r × v) (2.4) 1 The motion on magnetic surfaces of finite curvature received some attention in recent years both in the classical [11][12][13][14] and the quantum treatment [11,15,12,16,17]. One motivation for introducing a non-vanishing curvature is the possibility to study the quantum spectrum of the free magnetic motion on a compact domain (a modular domain in the case of constant negative curvature).…”

Section: Classical Motionmentioning

confidence: 99%

Magnetic edge states

Hornberger

Smilansky

2002

Physics Reports

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Magnetic edge states are responsible for various phenomena of magneto-transport. Their importance is due to the fact that, unlike the bulk of the eigenstates in a magnetic system, they carry electric current along the boundary of a confined domain. Edge states can exist both as interior (quantum dot) and exterior (anti-dot) states. In the present report we develop a consistent and practical spectral theory for the edge states encountered in magnetic billiards. It provides an objective definition for the notion of edge states, is applicable for interior and exterior problems, facilitates efficient quantization schemes, and forms a convenient starting point for both the semiclassical description and the statistical analysis. After elaborating these topics we use the semiclassical spectral theory to uncover nontrivial spectral correlations between the interior and the exterior edge states. We show that they are the quantum manifestation of a classical duality between the trajectories in an interior and an exterior magnetic billiard.Comment: 170 pages, 48 figures (high quality version available at http://www.klaus-hornberger.de

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“…7,16 There are several investigations of stochastization of motion in the magnetic billiards with and without curvature of the boundaries of domain where charged particles move. 18,23,43,44 One of possible triggers of stochastization of motion is the destruction of adiabatic invariance (for magnetic billiards, the role of this invariant is played by the classical magnetic moment 35 ). It was shown that destruction of the adiabaticity in magnetic billiards can realize in case of an inhomogeneous magnetic field.…”

Section: Introductionmentioning

confidence: 99%

Violation of adiabaticity in magnetic billiards due to separatrix crossings

Artemyev

Neishtadt

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of the adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance.

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Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

Cited by 24 publications

References 7 publications

A survey on polynomial in momenta integrals for billiard problems

A survey on polynomial in momenta integrals for billiard problems

Magnetic edge states

Violation of adiabaticity in magnetic billiards due to separatrix crossings

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