A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

Oxford University Press in Africa, 1927â80

Manifolds all of whose geodesics are closed

Symplectic and Poisson geometry on b-manifolds

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 G-equivariant way, to the linearized foliation in a neighborhood of a compact singular nondegenerate orbit. We also show that the nondegeneracy condition is not equivalent to the nonresonance condition for smooth systems.

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Equivariant normal form for nondegenerate singular orbits of integrable hamiltonian systems

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9].

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Codimension one symplectic foliations and regular Poisson structures

We prove the action-angle theorem in the general, and most natural, context of integrable systems on Poisson manifolds, thereby generalizing the classical proof, which is given in the context of symplectic manifolds. The topological part of the proof parallels the proof of the symplectic case, but the rest of the proof is quite different, since we are naturally led to using the calculus of polyvector fields, rather than differential forms; in particular, we use in the end a Poisson version of the classical Caratheodory-Jacobi-Lie theorem, which we also prove. At the end of the article, we generalize the action-angle theorem to the setting of non-commutative integrable systems on Poisson manifolds.

https://hal.archives-ouvertes.fr/hal-00292398/document

Action-angle Coordinates for Integrable Systems on Poisson Manifolds

We study Hamiltonian actions on $b$-symplectic manifolds with a focus on the effective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classifies these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus.

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Eva Miranda

Papers

Symplectic and Poisson geometry on b-manifolds

Equivariant normal form for nondegenerate singular orbits of integrable hamiltonian systems

Codimension one symplectic foliations and regular Poisson structures

Action-angle Coordinates for Integrable Systems on Poisson Manifolds

Toric actions on b-symplectic manifolds