On semi-global invariants for focus–focus singularities

Ngọc, San Vũ

doi:10.1016/s0040-9383(01)00026-x

Cited by 75 publications

(70 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and H xy has the same form as before, see (11). Using the (ξ, η, p ξ , p η ) variables we obtain the following well known result, see Appendix A for a proof.…”

Section: Prelude: the 1:1 Resonancementioning

confidence: 55%

“…The amazing feature of (1) is that it is multi-valued and thus not differentiable at the origin. The multi-valuedness means that no matter how small the initial perturbation from the equilibrium is, one can always obtain all possible values for W. In general, such multi-valuedness has been described for integrable foliations near focus-focus points by Vũ Ngoc [11]. In some sense our result is a special case of his.…”

Section: Introductionmentioning

confidence: 67%

“…A system of generalized action-angle coordinates for the isotropic harmonic oscillator (11) is given by the action H xy with conjugate angle:…”

Section: Lemmamentioning

confidence: 99%

See 2 more Smart Citations

Monodromy in the resonant swing spring

Dullin

Giacobbe

Cushman

2004

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

In this paper, it is shown that an integrable approximation of the spring pendulum, when tuned to be in 1:1:2 resonance, has monodromy. The stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation number of the integrable approximation. Due to the monodromy, this rotation number is not a globally defined function of the integrals. In fact at lowest order it is given by arg(χ + iλ), where χ and λ are functions of the integrals. The resonant swing spring is therefore a system where monodromy has easily observed physical consequences

show abstract

“…and H xy has the same form as before, see (11). Using the (ξ, η, p ξ , p η ) variables we obtain the following well known result, see Appendix A for a proof.…”

Section: Prelude: the 1:1 Resonancementioning

confidence: 55%

Section: Introductionmentioning

confidence: 67%

See 1 more Smart Citation

Monodromy in the resonant swing spring

Dullin

Giacobbe

Cushman

2004

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…The two topological invariants defined above completely determine the topological type of a Lagrangian bundle, as mentioned in [Bat88,DD87,Dui80,Luk08,Ngo03,Zun03]. This can also be seen using arguments in equivariant obstruction theory which are akin to those used to define the universal Chern class.…”

Section: Monodromy Fix a Lagrangian Bundlementioning

confidence: 94%

Universal Lagrangian bundles

Sepe

2013

Geom Dedicata

View full text Add to dashboard Cite

Abstract. The obstruction to construct a Lagrangian bundle over a fixed integral affine manifold was constructed by Dazord and Delzant in [DD87] and shown to be given by 'twisted' cup products in [Sep11]. This paper uses the topology of universal Lagrangian bundles, which classify Lagrangian bundles topologically (cf. [Sep10b]), to reinterpret this obstruction as the vanishing of a differential on the second page of a Leray-Serre spectral sequence. Using this interpretation, it is shown that the obstruction of Dazord and Delzant depends on an important cohomological invariant of the integral affine structure on the base space, called the radiance obstruction, which was introduced by Goldman and Hirsch in [GH84]. Some examples, related to non-degenerate singularities of completely integrable Hamiltonian systems, are discussed.

show abstract

“…In our system, J 1 ¼ 2pm is defined globally and is single valued. However (as pointed out by Vũ Ngo_c [24,10]) the multivalued part of the second action J 2 ðm; EÞ near (0, 0) can be approximated as…”

Section: Monodromy Of the Circular Barrier Systemmentioning

confidence: 96%