Hamiltoniens, lagrangiens et sous-ensembles coïsotropes associés aux structures de Poisson

Abstract : In this thesis, we study cotangents paths. In chapter 1 we introduce for every Hamiltonian a Lagrangian on paths valued in the cotangent space whose stationary points projects onto Hamiltonian vector fields. We show that the remaining components of those stationary points tell whether the bivector field is Poisson or at least defines an integrable distribution - a class of bivector fields generalizing twisted Poisson structures that we study in detail. In chapter 2, we establish a local function version of a result due to Klimcík and Strobl then Cattaneo and Felder claiming that a bivector field on a manifold $M$ is Poisson if and only if cotangent paths form a coisotropic submabifold of the infinite dimensional symplectic manifold of paths valued in $T^*M$. Our purpose in chapter 2 is to prove this result without using the Banach manifold setting used by Cattaneo and Felder, which fails in the periodic case because cotangent loops do not form a Banach sub-manifold. Instead, we use local functions on the path space, a point of view that allows to speak of a coisotropic set
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Theses
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http://docnum.univ-lorraine.fr/prive/DDOC_T_2016_0079_TURKI.pdf
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https://hal.univ-lorraine.fr/tel-01752355
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Submitted on : Thursday, March 29, 2018 - 1:43:03 PM
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Yahya Turki. Hamiltoniens, lagrangiens et sous-ensembles coïsotropes associés aux structures de Poisson. Mathématiques générales [math.GM]. Université de Lorraine, 2016. Français. ⟨NNT : 2016LORR0079⟩. ⟨tel-01752355⟩

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