The Classical Field Theories
01 Jan 1960-Vol. 2, pp 226-858
About: The article was published on 1960-01-01. It has received 3018 citations till now. The article focuses on the topics: Classical unified field theories & Liouville field theory.
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TL;DR: In this paper, a mathematical framework is developed to study the mechanical behavior of material surfaces, and the tensorial nature of surface stress is established using the force and moment balance laws using a linear theory with non-vanishing residual stress.
Abstract: A mathematical framework is developed to study the mechanical behavior of material surfaces. The tensorial nature of surface stress is established using the force and moment balance laws. Bodies whose boundaries are material surfaces are discussed and the relation between surface and body stress examined. Elastic surfaces are defined and a linear theory with non-vanishing residual stress derived. The free-surface problem is posed within the linear theory and uniqueness of solution demonstrated. Predictions of the linear theory are noted and compared with the corresponding classical results. A note on frame-indifference and symmetry for material surfaces is appended.
2,641 citations
TL;DR: In this article, a generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role, and the theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin.
Abstract: A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the ${U}_{4}$ theory arises as a local gauge theory for the Poincar\'e group in space-time. The deviations of the ${U}_{4}$ theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.
2,421 citations
Cites background from "The Classical Field Theories"
...In 3See, for example, Eringen (1962), Jaunzemis (1967), Kro'ner (1964, 1968), Truesdell and Noll (1965), and Truesdell and Toupin (1960)....
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Book•
24 Feb 2012
TL;DR: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software.
Abstract: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.
2,372 citations
Cites methods from "The Classical Field Theories"
...We next consider a test problem from Turek (1996), which is a two-dimensional cylinder submerged into a fluid and surrounded by solid walls as illustrated in Figure 21....
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2,228 citations
TL;DR: The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body as mentioned in this paper, which can be expressed as follows: a body is regarded as a smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies.
Abstract: The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body. In a formal rational development of the subject, one first tries to state precisely what mathematical entities represent these physical concepts: a body is regarded to be a smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies1. Once these concepts are made precise one can proceed to the statement of general principles, such as the principle of objectivity or the law of balance of linear momentum, and to the statement of specific constitutive assumptions, such as the assertion that a force system can be resolved into body forces with a mass density and contact forces with a surface density, or the assertion that the contact forces at a material point depend on certain local properties of the configuration at the point. While the general principles are the same for all work in classical continuum mechanics, the constitutive assumptions vary with the application in mind and serve to define the material under consideration.
1,885 citations
Cites background from "The Classical Field Theories"
...The value 1 (F, ~7) of the response function I is a linear transformation over the nine-dimensional space of tensors, and the square brackets in (3.3) indicate that this transformation operates on the tensor L. 2 The response functions ~', va, T, 1 and ~ depend on the choice of the reference configuration. 1 See the sections of [ 4 ] cited above....
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...x For more extensive discussions of the foundations-of continuum mechanics see references [1]-- [ 4 ]....
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...1 A thorough discussion of these conservation laws is given in [ 4 ], w167 196--205,...
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References
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Book•
01 Jan 1954
TL;DR: Molecular theory of gases and liquids as mentioned in this paper, molecular theory of gas and liquids, Molecular theory of liquid and gas, molecular theories of gases, and liquid theory of liquids, مرکز
Abstract: Molecular theory of gases and liquids , Molecular theory of gases and liquids , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی
11,807 citations
Book•
01 Jan 1873
TL;DR: The most influential nineteenth-century scientist for twentieth-century physics, James Clerk Maxwell (1831-1879) demonstrated that electricity, magnetism and light are all manifestations of the same phenomenon: the electromagnetic field as discussed by the authors.
Abstract: Arguably the most influential nineteenth-century scientist for twentieth-century physics, James Clerk Maxwell (1831–1879) demonstrated that electricity, magnetism and light are all manifestations of the same phenomenon: the electromagnetic field. A fellow of Trinity College Cambridge, Maxwell became, in 1871, the first Cavendish Professor of Physics at Cambridge. His famous equations - a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density - first appeared in fully developed form in his 1873 Treatise on Electricity and Magnetism. This two-volume textbook brought together all the experimental and theoretical advances in the field of electricity and magnetism known at the time, and provided a methodical and graduated introduction to electromagnetic theory. Volume 2 covers magnetism and electromagnetism, including the electromagnetic theory of light, the theory of magnetic action on light, and the electric theory of magnetism.
9,565 citations
Book•
01 Jun 1984
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Abstract: Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of a complex symmetric matrix 4. The normal form of a complex skew-symmetric matrix 5. The normal form of a complex orthogonal matrix XII. Singular pencils of matrices: 1. Introduction 2. Regular pencils of matrices 3. Singular pencils. The reduction theorem 4. The canonical form of a singular pencil of matrices 5. The minimal indices of a pencil. Criterion for strong equivalence of pencils 6. Singular pencils of quadratic forms 7. Application to differential equations XIII. Matrices with non-negative elements: 1. General properties 2. Spectral properties of irreducible non-negative matrices 3. Reducible matrices 4. The normal form of a reducible matrix 5. Primitive and imprimitive matrices 6. Stochastic matrices 7. Limiting probabilities for a homogeneous Markov chain with a finite number of states 8. Totally non-negative matrices 9. Oscillatory matrices XIV. Applications of the theory of matrices to the investigation of systems of linear differential equations: 1. Systems of linear differential equations with variable coefficients. General concepts 2. Lyapunov transformations 3. Reducible systems 4. The canonical form of a reducible system. Erugin's theorem 5. The matricant 6. The multiplicative integral. The infinitesimal calculus of Volterra 7. Differential systems in a complex domain. General properties 8. The multiplicative integral in a complex domain 9. Isolated singular points 10. Regular singularities 11. Reducible analytic systems 12. Analytic functions of several matrices and their application to the investigation of differential systems. The papers of Lappo-Danilevskii XV. The problem of Routh-Hurwitz and related questions: 1. Introduction 2. Cauchy indices 3. Routh's algorithm 4. The singular case. Examples 5. Lyapunov's theorem 6. The theorem of Routh-Hurwitz 7. Orlando's formula 8. Singular cases in the Routh-Hurwitz theorem 9. The method of quadratic forms. Determination of the number of distinct real roots of a polynomial 10. Infinite Hankel matrices of finite rank 11. Determination of the index of an arbitrary rational fraction by the coefficients of numerator and denominator 12. Another proof of the Routh-Hurwitz theorem 13. Some supplements to the Routh-Hurwitz theorem. Stability criterion of Lienard and Chipart 14. Some properties of Hurwitz polynomials. Stieltjes' theorem. Representation of Hurwitz polynomials by continued fractions 15. Domain of stability. Markov parameters 16. Connection with the problem of moments 17. Theorems of Markov and Chebyshev 18. The generalized Routh-Hurwitz problem Bibliography Index.
9,334 citations