4,068 research outputs found
Completely integrable systems: a generalization
We present a slight generalization of the notion of completely integrable
systems to get them being integrable by quadratures. We use this generalization
to integrate dynamical systems on double Lie groups.Comment: Latex, 15 page
Lagrangian submanifolds and dynamics on Lie affgebroids
We introduce the notion of a symplectic Lie affgebroid and their Lagrangian
submanifolds in order to describe the Lagrangian (Hamiltonian) dynamics on a
Lie affgebroid in terms of this type of structures. Several examples are
discussed.Comment: 50 pages. Several sections update
On quasi-Jacobi and Jacobi-quasi bialgebroids
We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships
with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct
quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also
that the structures induced on their base manifolds are related via a quasi
Poissonization
Poisson-Jacobi reduction of homogeneous tensors
The notion of homogeneous tensors is discussed. We show that there is a
one-to-one correspondence between multivector fields on a manifold ,
homogeneous with respect to a vector field on , and first-order
polydifferential operators on a closed submanifold of codimension 1 such
that is transversal to . This correspondence relates the
Schouten-Nijenhuis bracket of multivector fields on to the Schouten-Jacobi
bracket of first-order polydifferential operators on and generalizes the
Poissonization of Jacobi manifolds. Actually, it can be viewed as a
super-Poissonization. This procedure of passing from a homogeneous multivector
field to a first-order polydifferential operator can be also understood as a
sort of reduction; in the standard case -- a half of a Poisson reduction. A
dual version of the above correspondence yields in particular the
correspondence between -homogeneous symplectic structures on and
contact structures on .Comment: 19 pages, minor corrections, final version to appear in J. Phys. A:
Math. Ge
A variational principle for volume-preserving dynamics
We provide a variational description of any Liouville (i.e. volume
preserving) autonomous vector fields on a smooth manifold. This is obtained via
a ``maximal degree'' variational principle; critical sections for this are
integral manifolds for the Liouville vector field. We work in coordinates and
provide explicit formulae
Integration of Dirac-Jacobi structures
We study precontact groupoids whose infinitesimal counterparts are
Dirac-Jacobi structures. These geometric objects generalize contact groupoids.
We also explain the relationship between precontact groupoids and homogeneous
presymplectic groupoids. Finally, we present some examples of precontact
groupoids.Comment: 10 pages. Brief changes in the introduction. References update
Hubbard Models as Fusion Products of Free Fermions
A class of recently introduced su(n) `free-fermion' models has recently been
used to construct generalized Hubbard models. I derive an algebra defining the
`free-fermion' models and give new classes of solutions. I then introduce a
conjugation matrix and give a new and simple proof of the corresponding
decorated Yang-Baxter equation. This provides the algebraic tools required to
couple in an integrable way two copies of free-fermion models. Complete
integrability of the resulting Hubbard-like models is shown by exhibiting their
L and R matrices. Local symmetries of the models are discussed. The
diagonalization of the free-fermion models is carried out using the algebraic
Bethe Ansatz.Comment: 14 pages, LaTeX. Minor modification
A general framework for nonholonomic mechanics: Nonholonomic Systems on Lie affgebroids
This paper presents a geometric description of Lagrangian and Hamiltonian
systems on Lie affgebroids subject to affine nonholonomic constraints. We
define the notion of nonholonomically constrained system, and characterize
regularity conditions that guarantee that the dynamics of the system can be
obtained as a suitable projection of the unconstrained dynamics. It is shown
that one can define an almost aff-Poisson bracket on the constraint AV-bundle,
which plays a prominent role in the description of nonholonomic dynamics.
Moreover, these developments give a general description of nonholonomic systems
and the unified treatment permits to study nonholonomic systems after or before
reduction in the same framework. Also, it is not necessary to distinguish
between linear or affine constraints and the methods are valid for explicitly
time-dependent systems.Comment: 50 page
- …