Configuration Bundle Research Articles

Gauge reduction in covariant field theory

In this work, we develop a Lagrangian reduction theory for covariant field theories with gauge symmetries. These symmetries are modeled by a Lie group fiber bundle acting fiberwisely on a configuration bundle. In order to reduce the variational principle, we utilize generalized principal connections, a type of Ehresmann connections that are equivariant by the fiberwise action. After obtaining the reduced equations, we give the reconstruction condition and we relate the vertical reduced equation with the Noether theorem. Lastly, we illustrate the theory with several examples, including the classical case (Lagrange–Poincaré reduction), Electromagnetism, symmetry-breaking and non-Abelian gauge theories.

A new canonical affine bracket formulation of Hamiltonian classical field theories of first order

It has been a long standing question how to extend, in the finite-dimensional setting, the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang–Mills theory.

A variational framework for higher order perturbations

A covariant, global, variational framework for perturbations in field theories is presented. Perturbations are obtained as vertical vector fields on the configuration bundle and they drag, exactly, solution into solutions. The flow of a perturbation drags solutions into solutions and the dragged perturbed solutions can be expanded in a series with respect to the flow parameter, hence it contains perturbations at any order. Mechanics is included as a special case. As a simple application, we recover the well-known discussion about stability of geodesics on a sphere [Formula: see text].

The dressing field method in gauge theories - geometric approach

<abstract><p>Recently, T. Masson, J. Francois, S. Lazzarini, C. Fournel and J. Attard have introduced a new method of the reduction of gauge symmetries called the dressing field method. In this paper we analyse this method from the fiber bundle point of view and we show the geometric implications for a principal bundle underlying a given gauge theory.We show how the existence of a dressing field satisfying certain conditions naturally leads to the reduction of the principal bundle and, as a consequence, to the reduction of the configuration and phase bundle of the system.</p></abstract>

Frölicher-smooth geometries, quantum jet bundles and BRST symmetry

We attempt a clarification of geometric aspects of quantum field theory by using the notion of smoothness introduced by Fr\"olicher and exploited by several authors in the study of functional bundles. A discussion of momentum and position representations in curved spacetime, in terms of generalized semi-densities, leads to a definition of quantum configuration bundle which is suitable for a treatment of that kind. A consistent approach to Lagrangian field theories, vertical infinitesimal symmetries and related currents is then developed, and applied to a formulation of BRST symmetry in a gauge theory of the Yang-Mills type.

MULTISYMPLECTIC GEOMETRY AND k-COSYMPLECTIC STRUCTURE FOR THE FIELD THEORIES AND THE RELATIVISTIC MECHANICS

The aim of this work is twofold: First, we extend the multisymplectic geometry already done for field theories to the relativistic mechanics by introducing an appropriate configuration bundle. In particular, we developed the model to obtain the Hamilton–De Donder–Weyl equations to the movement of a relativistic charged particle immerged in an electromagnetic field. Second, we have found a direct relationship between the multisymplectic geometry and the k-cosymplectic structure of a physical system.

A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

AbstractA theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchhoff's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null‐space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates first integrals of the equilibrium equations to Lie group actions on the configuration bundle, so‐called symmetries. The symmetries relevant for rod mechanics are frame‐indifference, isotropy, and uniformity. We show that a completely analogous and self‐contained theory of discrete rods can be formulated in which the arc‐length is a discrete variable ab initio. In this formulation, the potential energy density is defined directly on pairs of points along the arc‐length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identifies exact first integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application. Copyright © 2010 John Wiley & Sons, Ltd.

Jet methods in time-dependent Lagrangian biomechanics

Abstract In this paper we propose the time-dependent generalization of an ‘ordinary’ autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total mass-inertia matrix of the human body segments. This is the base manifold for standard autonomous biomechanics. To make its time-dependent generalization, we need to extend it with a real time axis. By this extension, using techniques from fibre bundles, we defined the biomechanical configuration bundle. On the biomechanical bundle we define vector-fields, differential forms and affine connections, as well as the associated jet manifolds. Using the formalism of jet manifolds of velocities and accelerations, we develop the time-dependent Lagrangian biomechanics. Its underlying geometric evolution is given by the Ricci flow equation.

Covariant formulation of the governing equations of continuum mechanics in an Eulerian description

We present the balance relations for a continuum in the Eulerian formulation in a pure covariant fashion. Based on the analysis of nonrelativistic particle mechanics, we adapt the covariant description to the case of a continuum. The use of the covariant Nijenhuis differential as well as the splitting of the vertical configuration bundle are the key objects that allow a coordinate-free representation. We state the balance equations such that they are valid, also when time variant transformations are applied, which leads to a nontrivial space-time connection and a metric which explicitly depends on the time.

Nonstandard connections in k-cosympletic field theory

In the jet-bundle description of time-dependent mechanics there are some elements, such as the Lagrangian energy and the construction of the Hamiltonian formalism, which require the prior choice of a connection. This situation is analyzed by Echeverría-Enríquez et al. [J. Phys. A 28, 5553–5567 (1995)]. The aim of this paper is to extend the results in that paper to first order field theory, using the k-cosymplectic formalism described by de León and co-workers [J. Math. Phys. 39, 876–893 (1998); 42, 2092–2104 (2001)]. If the trivial configuration bundle of a Lagrangian system is endowed with one connection, different from the trivial one given by the product structure, we study the consequences on the geometric elements of the theory, the dynamical equations and the variational principles.

Time-dependent Lagrangians invariant by a vector field

We study the reduction of non-autonomous regular Lagrangian systems by symmetries, which are generated by vector fields associated with connections in the configuration bundle of the system $Q\times\real\to\real$. These kind of symmetries generalize the idea of ``time-invariance'' (which corresponds to taking the trivial connection in the above trivial bundle).

On the geodesic form of second-order dynamic equations

It is shown that any second-order dynamic equation on a configuration bundle Q→R of nonrelativistic mechanics is equivalent to a geodesic equation with respect to a (nonlinear) connection on the tangent bundle TQ→Q. The case of quadratic dynamic equations is analyzed in detail. The equation for Jacobi vector fields is constructed and investigated by the geometric methods.

Nonholonomic constraints in time-dependent mechanics

The constraint reaction force of ideal nonholonomic constraints in time-dependent mechanics on a configuration bundle Q→R is obtained. Using the vertical extension of Hamiltonian formalism to the vertical tangent bundle VQ of Q→R, the Hamiltonian of a nonholonomic constrained system is constructed. The present setting is more general than the one usually considered in the literature on nonholonomic mechanics.

The Second Noether Theorem in the formalism of jet-bundles: Symmetries and degeneration

By means of the jet-bundle formalism, the Second Noether Theorem is formulated for a general first-order Lagrangian field theory with infinitesimal local symmetries. These symmetries are implemented by a linear differential operator acting between the sections of a vector bundle and vector fields on the configuration bundle. The problem of the degeneration of the Lagrangian system is examined from a covariant and an instantaneous (i.e. space+time split) viewpoint. It is shown that in the instantaneous approach the presence of infinitesimal local symmetries leads to degeneration of the theory. Vertical local symmetries are shown to imply degeneration also in the covariant formalism. These results can be extended to higher-order Lagrangians as well.