Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance

We recognize some universal features of macroscopic dynamics describing the approach of a well-established level of description (that is, successfully tested by experimental observations) to equilibrium. The universal features are collected in a general equation for the nonequilibrium reversible-irreversible coupling (abbreviated as GENERIC). In this paper we formulate a GENERIC, derive properties of its solutions, and discuss their physical interpretation. The relation of the GENERIC with thermodynamics is most clearly displayed in a formulation that uses contact structures. The GENERIC is also discussed in the presence of noise. In applications we either search for new governing equations expressing our insight into a particular complex fluid or take well-established governing equations and cast them into the form of a GENERIC. In the former case we obtain the governing equations as particular realizations of the GENERIC structure; in the latter case we justify the universality of the GENERIC and derive some properties of solutions. Both types of applications are discussed mainly in the following paper [Phys. Rev. E 56, 6633 (1997)].

Dynamics and thermodynamics of complex fluids. I. Development of a general formalism

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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Finite elements in computational electromagnetism

For a number of well-known time-evolution equations for nonequilibrium systems we extract a common structure from these equations, referred to as a general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). This fundamental structure is determined by four building blocks, two ``potentials'' (total energy and entropy) and two ``matrices.'' We illustrate for various examples how three of the four building blocks can be determined in a rather straightforward manner so that, within our GENERIC approach to nonequilibrium dynamics, understanding of a given nonequilibrium system is reduced to determining a single ``metric matrix,'' or friction matrix, either empirically or by more microscopic considerations. In formulating nonisothermal polymer kinetic theories, we show how the general structure provides a clear distinction between spring potentials of energetic and entropic origins in the various time-evolution equations.

Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism

It has been recognised that numerical computations of magnetic fields by the finite-element method may require new types of elements, whose degrees of freedom are not field values at mesh nodes, but other field-related quantities like e.g. circulations along edges of the mesh. A rationale for the use of these special ‘mixed’ elements can be obtained if one cxpresses basic equations in terms of differential forms, instead of vector fields. The paper gives an elementary introduction to this point of view, presents Whitney forms (the mixed finite elements alluded to), and sketches two numerical methods (dual, in some sense), for eddy-current studies, based on these elements.

Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.

Applied Differential Geometry

Preface Glossary of notation Introduction 1. Tensors in linear spaces 2. Manifolds 3. Transformations 4. The calculus of differential forms 5. Applications of the exterior calculus 6. Classical electrodynamics 7. Dynamics of particles and fields 8. Calculus on fiber bundles 9. Gravitation Bibliography Index.

William L. Burke

Papers

Applied Differential Geometry

Applied Differential Geometry.

Spacetime, geometry, cosmology

The coupling of gravitational radiation to nonrelativistic sources