Marco Castrillón López

Some remarks on Lagrangian and Poisson reduction for field theories

Given a Hamiltonian system on a fiber bundle, the Poisson covariant formulation of the Hamilton equations is described. When the fiber bundle is a G-principal bundle and the Hamiltonian density is G-invariant, the reduction of this formulation is studied thus obtaining the analog of the Lie-Poisson reduction for field theories. The relation of this reduction with the Lagrangian reduction and the Lagrangian and Poisson reduction for electromagnetism are also analyzed.

Reduction in principal fiber bundles: covariant Euler-Poincaré equations.

In Lagrangian mechanics the simplest example of reduction is the Euler-Poincare reduction. In this case, the configuration space is a Lie group G and the Lagrangian function L:TG→R is invariant under the natural action of the group on TG by right translations. Then L induces a function l:(TG)/G≅g→R, g being the Lie algebra of G, and the Euler-Lagrange equations defined by L transform into a new group of equations for l called the Euler-Poincare equations. For example, this is the case for the dynamics of the rigid body. In the present paper, the authors extend the idea of reduction to Lagrangian field theory. In this framework, the analogous configuration is a principal bundle π:P→M with structure group G (a matrix group) and a Lagrangian L:J1P→R invariant under the natural action of G on the 1-jet bundle defined by (j1xs)⋅g=j1x(Rg∘s), where Rg denotes the right translation by g on P. Let l:(J1P)/G→R be the induced mapping. It is proved that the Euler-Lagrange equations define a group of equations for critical sections, which generalize the Euler-Poincare equations of mechanics. As is well known, the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M. This fact gives a geometrical meaning to the previous reduction. In particular, the critical sections of the Euler-Poincare equations are sections of this bundle, and therefore they can be understood as principal connections of π:P→M. The authors exploit this idea in order to explain the compatibility conditions needed for reconstruction. The Euler-Poincare equations do not suffice to reconstruct the Euler-Lagrange equations. Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections. This fact is characteristic of field theory and does not appear in classical mechanics. Finally, the authors study the Euler-Poincare equations in two examples from the variational approach to harmonic mappings.

Differential characters and cohomology of the moduli of flat connections

Let

LR property of non-well-formed scales

\pi\colon P\to M

On the Step-Patterns of Generated Scales that are Not Well-Formed

be a principal bundle and

Un-reduction in field theory

p

Morse Families and Lagrangian Submanifolds

an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define an homology map

Euler–Poincaré Reduction by a Subgroup of Symmetries as an Optimal Control Problem

\chi^{k} : H_{2r-k-1}(M)\times H_{k}(\mathcal{F}/\mathcal{G})\to \mathbb{R}/\mathbb{Z}

Lie algebra pairing and the Lagrangian and Hamiltonian equations in gauge-invariant problems

, for

Structure symplectique généralisée sur le fibre des connexions

k<r-1