On global action‐angle coordinates

Duistermaat, J. J.

doi:10.1002/cpa.3160330602

Cited by 472 publications

(619 citation statements)

References 2 publications

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“…After describing the features of the classical dynamics in qualitative terms and explaining the relation to the mean-field phase diagram, we in-troduce some ideas from the theory of classical integrable systems, notably action-angle coordinates, that are then applied to the system of interest. It is in this section that we meet the phenomenon of Hamiltonian monodromy, a topological obstruction to the global existence of actionangle coordinates (Section III D) [7][8][9]. This material may be unfamiliar to many readers, so we have tried to be pedagogical in our presentation (other introductions suitable for physicists may be found in the appendices to Ref.…”

Section: Introductionmentioning

confidence: 99%

Spin-1 microcondensate in a magnetic field

Lamacraft

2011

Phys. Rev. A

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We study a spin 1 Bose condensate small enough to be treated as a single magnetic 'domain': a system that we term a microcondensate. Because all particles occupy a single spatial mode, this quantum many body system has a well defined classical limit consisting of three degrees of freedom, corresponding to the three macroscopically occupied spin states. We study both the classical limit and its quantization, finding an integrable system in both cases. Depending on the sign of the ratio of the spin interaction energy and the quadratic Zeeman energy, the classical limit displays either a separartrix in phase space, or Hamiltonian monodromy corresponding to non-trivial phase space topology. We discuss the quantum signatures of these classical phenomena using semiclassical quantization as well as an exact solution using the Bethe ansatz.

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Section: Introductionmentioning

confidence: 99%

Spin-1 microcondensate in a magnetic field

Lamacraft

2011

Phys. Rev. A

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“…This fibre is S 1 × I 1 , where I 1 is a Kodaira type I 1 fibre (a pinched torus). So, fibres of type (2,2) are singular along a circle. Fibres of type (2,2) lie over the edges of Γ; • type (1,2).…”

Section: Introductionmentioning

confidence: 99%

Symplectic invariants of some families of Lagrangian $T^3$-fibrations

Bernard¹

2004

Journal of Symplectic Geometry

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We construct families of Lagrangian 3-torus fibrations resembling the topology of some of the singularities in Topological Mirror Symmetry [8]. We perform a detailed analysis of the affine structure on the base of these fibrations near their discriminant loci. This permits us to classify the aforementioned families up to fibre preserving symplectomorphism. The kind of degenerations we investigate give rise to a large number of symplectic invariants.

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“…Roughly speaking, the flip effect can be viewed as a trajectory in a neighborhood of the separatrix, which goes from a point close to an unstable state to a point near another unstable state. We can make the same analysis for the first equation of (8). Using Eq.…”

Section: General Description Of the Evolution Of The Systemmentioning

confidence: 99%

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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On global action‐angle coordinates

Cited by 472 publications

References 2 publications

Spin-1 microcondensate in a magnetic field

Spin-1 microcondensate in a magnetic field

Symplectic invariants of some families of Lagrangian $T^3$-fibrations

The tennis racket effect in a three-dimensional rigid body

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