Quantifying air–water turbulence with moment field equations

Electromagnetic Radiation is a graduate level book on classical electrodynamics with a strong emphasis on radiation. This book is meant to quickly and efficiently introduce students to the electromagnetic radiation science essential to a practicing physicist. While a major focus is on light and its interactions, topics in radio frequency radiation, x-rays, and beyond are also treated. Special emphasis is placed on applications, with many exercises and homework problems. The format of the book is designed to convey the basic concepts of a topic in the main central text in the book in a mathematically rigorous manner, but with detailed derivations routinely relegated to the accompanying side notes or end of chapter “Discussions.” The book is composed of four parts: Part I is a review of basic E&M, and assumes the reader has a had a good upper division undergraduate course, and while it offers a concise review of topics covered in such a course, it does not treat any given topic in detail; specifically electro- and magnetostatics. Part II addresses the origins of radiation in terms of time variations of charge and current densities within the source, and presents Jefimenko’s field equations as derived from retarded potentials. Part III introduces special relativity and its deep connection to Maxwell’s equations, together with an introduction to relativistic field theory, as well as the relativistic treatment of radiation from an arbitrarily accelerating charge. A highlight of this part is a chapter on the still partially unresolved problem of radiation reaction on an accelerating charge. Part IV treats the practical problems of electromagnetic radiation interacting with matter, with chapters on energy transport, scattering, diffraction and finally an illuminating, application-oriented treatment of fields in confined environments.

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General Relativity

10.1093/oso/9780198822158.001.0001 ◽

2019 ◽

Cited By ~ 1

Author(s):

Steven Carlip

Keyword(s):

General Relativity ◽

High Energy Physics ◽

Hamiltonian Formulation ◽

Experimental Tests ◽

High Energy ◽

Gravitational Energy ◽

Field Equations ◽

Physics First ◽

Condensed Matter Theory ◽

Introductory Textbooks

This work is a short textbook on general relativity and gravitation, aimed at readers with a broad range of interests in physics, from cosmology to gravitational radiation to high energy physics to condensed matter theory. It is an introductory text, but it has also been written as a jumping-off point for readers who plan to study more specialized topics. As a textbook, it is designed to be usable in a one-quarter course (about 25 hours of instruction), and should be suitable for both graduate students and advanced undergraduates. The pedagogical approach is “physics first”: readers move very quickly to the calculation of observational predictions, and only return to the mathematical foundations after the physics is established. The book is mathematically correct—even nonspecialists need to know some differential geometry to be able to read papers—but informal. In addition to the “standard” topics covered by most introductory textbooks, it contains short introductions to more advanced topics: for instance, why field equations are second order, how to treat gravitational energy, what is required for a Hamiltonian formulation of general relativity. A concluding chapter discusses directions for further study, from mathematical relativity to experimental tests to quantum gravity.

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Field equations in the neighbourhood of a particle in a conformal theory of gravitation. II

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1969.0180 ◽

1969 ◽

Vol 313 (1512) ◽

pp. 71-82 ◽

Cited By ~ 2

Keyword(s):

Local Geometry ◽

Field Equations ◽

Schwarzschild Solution ◽

Conformal Theory ◽

Conformally Invariant ◽

Coordinate Singularity ◽

Previous Solution ◽

Theory Of Gravitation ◽

Spherically Symmetric Metric ◽

Radial Variable

The field equations in the neighbourhood of a particle for a spherically symmetric metric in the conformal theory of gravitation put forward by Hoyle & Narlikar are examined. As the theory is conformally invariant, one can use different but physically equivalent conformal frames to study the equations. Previously these equations were studied in a conformal frame which, though suitable far away from the isolated particle, turns out not to be suitable in the neighbourhood of the particle. In the present paper a solution in a conformal frame is obtained that is suitable for considering regions near the particle. The solution thus obtained differs from the previous one in several respects. For example, it has no coordinate singularity for any non-zero value of the radial variable, unlike the previous solution or the Schwarzschild solution. It is also shown with the use of this solution that in this theory distant matter has an effect on local geometry.

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Phase-field gradient theory

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/s00033-020-01441-2 ◽

2021 ◽

Vol 72 (2) ◽

Author(s):

Luis Espath ◽

Victor Calo

Keyword(s):

Free Energy ◽

Field Equation ◽

Field Gradient ◽

Phase Field ◽

Constitutive Relations ◽

The Body ◽

Second Grade ◽

Gradient Theory ◽

Boundary Edge ◽

Field Equations

AbstractWe propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

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