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Toric structures on near-symplectic 4-manifolds

David T. Gay, Margaret Symington (2009)

Journal of the European Mathematical Society

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of...

Two-input control systems on the euclidean group  SE (2)

Ross M. Adams, Rory Biggs, Claudiu C. Remsing (2013)

ESAIM: Control, Optimisation and Calculus of Variations

Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis...

Une caractérisation des formes symplectiques

Bruno Sévennec (1998)

Annales de l'institut Fourier

On montre qu’une 2-forme non nulle sur une variété M , telle que le pseudogroupe des difféomorphismes locaux la préservant soit transitif sur le fibré des directions tangentes, est symplectique si la dimension de M n’est pas 6 . De plus, il y a un contre-exemple en dimension 6, dont on montre qu’il est essentiellement unique.

Universal lifting theorem and quasi-Poisson groupoids

David Inglesias-Ponte, Camille Laurent-Gengoux, Ping Xu (2012)

Journal of the European Mathematical Society

We prove the universal lifting theorem: for an α -simply connected and α -connected Lie groupoid Γ with Lie algebroid A , the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on Γ . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular,...

Universal prolongation of linear partial differential equations on filtered manifolds

Katharina Neusser (2009)

Archivum Mathematicum

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that the solution space of such equations is always finite dimensional.

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Liana David (0)

Annales de l’institut Fourier

Variational principles and symmetries on fibered multisymplectic manifolds

Jordi Gaset, Pedro D. Prieto-Martínez, Narciso Román-Roy (2016)

Communications in Mathematics

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multisymplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special...

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