Global Aspects of Classical Integrable Systems

Cushman, Richard; Bates, Larry

doi:10.1007/978-3-0348-8891-2

Cited by 274 publications

(287 citation statements)

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“…The tennis racket effect was initially mathematically analyzed in [12], and reproduced in [5]. In this paper, we expand considerably this initial treatment by deriving exact and asymptotic formulae describing the flip of the head of the racket.…”

Section: Discussionmentioning

confidence: 99%

“…In the case of integrable systems with few degrees of freedom [3,4], an efficient approach is based on a geometric analysis to characterize the dynamical properties of the mechanical system [5,6]. Such geometric phenomena are typically at the origin of the robustness of particular effects which can be observed experimentally.…”

Section: Introductionmentioning

confidence: 99%

“…In spite of the apparently well-known character of TRE, this property is not discussed in standard text-books of classical mechanics [1,3,4]. To the best of our knowledge, the only mathematical study of this geometric effect has been made in [12] and reproduced in [5]. However, this work only focuses on a specific case where the moments of inertia satisfy a constraint of the form I 1 + I 2 ≃ I 3 .…”

Section: Introductionmentioning

confidence: 99%

“…Note that the Montgomery phase also occurs in the free rotation of a rigid body when the angular momentum undergoes a cyclic trajectory. The goal of this paper is therefore to extend the preliminary analysis made in [5,12] in different directions. We choose a suitable set of Euler angles which allows us to give a complete and unified analytical and geometrical description of TRE.…”

Section: Introductionmentioning

confidence: 99%

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The tennis racket effect in a three-dimensional rigid body

Damme¹,

Mardešić²,

Sugny³

2017

Physica D: Nonlinear Phenomena

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We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a three-dimensional rigid body. This effect is characterized by a flip (π-rotation) of the head of the racket when a full (2π) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Different examples are discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The tennis racket effect in a three-dimensional rigid body

Damme¹,

Mardešić²,

Sugny³

2017

Physica D: Nonlinear Phenomena

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“…However, little has been done in order to provide a theoretical description of the phenomenon of polarization attraction. Our aim in this article is to make a step in this direction by introducing a new set of tools that find their origin in recently developed mathematical techniques, in particular for the study of Hamiltonian singularities [28,29]. We have successfully used this mathematical approach in Refs.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian tools for the analysis of optical polarization control

et al. 2012

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The study of the polarization dynamics of two counterpropagating beams in optical fibers has recently been the subject of a growing renewed interest, from both the theoretical and experimental points of view. This system exhibits a phenomenon of polarization attraction, which can be used to achieve a complete polarization of an initially unpolarized signal beam, almost without any loss of energy. Along the same way, an arbitrary polarization state of the signal beam can be controlled and converted into any other desired state of polarization, by adjusting the polarization state of the counterpropagating pump beam. These properties have been demonstrated in various different types of optical fibers, i.e., isotropic fibers, spun fibers, and telecommunication optical fibers. This article is aimed at providing a rather complete understanding of this phenomenon of polarization attraction on the basis of new mathematical techniques recently developed for the study of Hamiltonian singularities. In particular, we show the essential role that play the peculiar topological properties of singular tori in the process of polarization attraction. We provide here a pedagogical introduction to this geometric approach of Hamiltonian singularities and give a unified description of the polarization attraction phenomenon in various types of optical fiber systems.

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Quantum Monodromyand Molecular Spectroscopy

Child

2007

Advances in Chemical Physics

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Global Aspects of Classical Integrable Systems

Cited by 274 publications

References 0 publications

The tennis racket effect in a three-dimensional rigid body

The tennis racket effect in a three-dimensional rigid body

Hamiltonian tools for the analysis of optical polarization control

Quantum Monodromyand Molecular Spectroscopy

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