Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case

Eliasson, Lennart

doi:10.1007/bf02566590

Cited by 222 publications

(254 citation statements)

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“…1) In the case when O is a point and G is trivial, the above theorem is due to Vey [22] in the analytic case, and Eliasson [8,9] in the smooth case. The smooth one-degree-of freedom case is due to Colin de Verdière and Vey [3].…”

Section: 2mentioning

confidence: 99%

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Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems

Miranda

Zung

2004

Annales Scientifiques de l’École Normale Supérieure

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Abstract. We consider an integrable Hamiltonian system with n-degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a Gequivariant way, to the linearized foliation in a neighborhood of a compact singular non-degenerate orbit. We also show that the non-degeneracy condition is not equivalent to the non-resonance condition for smooth systems.Résumé. On considère un système hamiltonien intégrableà n degrés de liberté et une action symplectique d'un groupe de Lie compact G qui laisse invariantes les intégrales premières. On prouve que le feuilletage lagrangien singulier attachéà ce système hamiltonien est symplectiquementéquivalent, de façon Gequivariante, au feuilletage linearisé dans un voisinage d'une orbite compacte singulière. On démontre aussi que la condition de non-dégénéréscence n'est paséquivalenteà la non-résonance pour les systèmes différentiables.

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Section: 2mentioning

confidence: 99%

“…The results of Eliasson [8,9] (for smooth systems) and Vey [22] (for real analytic systems) show that there is a fibration-preserving symplectomorphism from a neighborhood of O in (M 2n , ω, F) to a neighborhood of the origin of the linear system…”

Section: The Case Of a Fixed Pointmentioning

confidence: 99%

Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems

Miranda

Zung

2004

Annales Scientifiques de l’École Normale Supérieure

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“…Eliasson [3], Tien-Zung and Miranda [17]). Generically, the function components of the fibration-i.e., the integrals of the system -can be reduced to quadratic polynomials.…”

Section: The Family L(1 2)mentioning

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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“…The case of a nonresonant singular point as considered here is more involved. It has been shown by H. Rüssman [Ru1] for n = 2 and in general by J. Vey [Ve] and H. Ito [It] that the existence of n first integrals in involution forces the convergence of the normalization to Birkhoff normal form (H. Eliasson settled the analogue of Vey's theorem in the C ∞ case [El1], [El3]). L. Stolovitch gave a unified approach to Bruno's theorem cited before and Vey's and Ito's theorems ([St1], [St2]).…”

Section: Introductionmentioning

confidence: 99%