112 research outputs found
Poisson actions up to homotopy and their quantization
Symmetries of Poisson manifolds are in general quantized just to symmetries
up to homotopy of the quantized algebra of functions. It is therefore
interesting to study symmetries up to homotopy of Poisson manifolds. We notice
that they are equivalent to Poisson principal bundles and describe their
quantization to symmetries up to homotopy of the quantized algebras of
functions.Comment: 8 page
Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system
The Mishchenko-Fomenko theorem on superintegrable Hamiltonian systems is
generalized to superintegrable Hamiltonian systems with noncompact invariant
submanifolds. It is formulated in the case of globally superintegrable
Hamiltonian systems which admit global generalized action-angle coordinates.
The well known Kepler system falls into two different globally superintegrable
systems with compact and noncompact invariant submanifolds.Comment: 23 page
Jacobi Structures in
The most general Jacobi brackets in are constructed after
solving the equations imposed by the Jacobi identity. Two classes of Jacobi
brackets were identified, according to the rank of the Jacobi structures. The
associated Hamiltonian vector fields are also constructed
Generalized n-Poisson brackets on a symplectic manifold
On a symplectic manifold a family of generalized Poisson brackets associated
with powers of the symplectic form is studied. The extreme cases are related to
the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can
be obtained in a similar way.Comment: Latex, 10 pages, to appear in Mod. Phys. Lett.
Poisson sigma models and symplectic groupoids
We consider the Poisson sigma model associated to a Poisson manifold. The
perturbative quantization of this model yields the Kontsevich star product
formula. We study here the classical model in the Hamiltonian formalism. The
phase space is the space of leaves of a Hamiltonian foliation and has a natural
groupoid structure. If it is a manifold then it is a symplectic groupoid for
the given Poisson manifold. We study various families of examples. In
particular, a global symplectic groupoid for a general class of two-dimensional
Poisson domains is constructed.Comment: 34 page
Coupling Poisson and Jacobi structures on foliated manifolds
Let M be a differentiable manifold endowed with a foliation F. A Poisson
structure P on M is F-coupling if the image of the annihilator of TF by the
sharp-morphism defined by P is a normal bundle of the foliation F. This notion
extends Sternberg's coupling symplectic form of a particle in a Yang-Mills
field. In the present paper we extend Vorobiev's theory of coupling Poisson
structures from fiber bundles to foliations and give simpler proofs of
Vorobiev's existence and equivalence theorems of coupling Poisson structures on
duals of kernels of transitive Lie algebroids over symplectic manifolds. Then
we discuss the extension of the coupling condition to Jacobi structures on
foliated manifolds.Comment: LateX, 38 page
Nonholonomic systems with symmetry allowing a conformally symplectic reduction
Non-holonomic mechanical systems can be described by a degenerate
almost-Poisson structure (dropping the Jacobi identity) in the constrained
space. If enough symmetries transversal to the constraints are present, the
system reduces to a nondegenerate almost-Poisson structure on a ``compressed''
space. Here we show, in the simplest non-holonomic systems, that in favorable
circumnstances the compressed system is conformally symplectic, although the
``non-compressed'' constrained system never admits a Jacobi structure (in the
sense of Marle et al.).Comment: 8 pages. A slight edition of the version to appear in Proceedings of
HAMSYS 200
Geometrical aspects of integrable systems
We review some basic theorems on integrability of Hamiltonian systems, namely
the Liouville-Arnold theorem on complete integrability, the Nekhoroshev theorem
on partial integrability and the Mishchenko-Fomenko theorem on noncommutative
integrability, and for each of them we give a version suitable for the
noncompact case. We give a possible global version of the previous local
results, under certain topological hypotheses on the base space. It turns out
that locally affine structures arise naturally in this setting.Comment: It will appear on International Journal of Geometric Methods in
Modern Physics vol.5 n.3 (May 2008) issu
Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds
The obstruction to the existence of global action-angle coordinates of
Abelian and noncommutative (non-Abelian) completely integrable systems with
compact invariant submanifolds has been studied. We extend this analysis to the
case of noncompact invariant submanifolds.Comment: 13 pages, to be published in J. Math. Phys. (2007
Poisson Geometry in Constrained Systems
Constrained Hamiltonian systems fall into the realm of presymplectic
geometry. We show, however, that also Poisson geometry is of use in this
context.
For the case that the constraints form a closed algebra, there are two
natural Poisson manifolds associated to the system, forming a symplectic dual
pair with respect to the original, unconstrained phase space. We provide
sufficient conditions so that the reduced phase space of the constrained system
may be identified with a symplectic leaf in one of those. In the second class
case the original constrained system may be reformulated equivalently as an
abelian first class system in an extended phase space by these methods.
Inspired by the relation of the Dirac bracket of a general second class
constrained system to the original unconstrained phase space, we address the
question of whether a regular Poisson manifold permits a leafwise symplectic
embedding into a symplectic manifold. Necessary and sufficient for this is the
vanishing of the characteristic form-class of the Poisson tensor, a certain
element of the third relative cohomology.Comment: 41 pages, more detailed abstract in paper; v2: minor corrections and
an additional referenc
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