Differential Forms, with Applications to the Physical Sciences.

Linton, F. E. J.; Flanders, Harley

doi:10.2307/2311963

Cited by 82 publications

(34 citation statements)

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“…1 From a philosophical point of view, the co-moving frame shall not be called an inertial frame since we cannot distinguish between the forces due to the gravitational forces and due to the acceleration (inertial). 2 Often, its study is conducted by using differential forms [66]. We will not make much use of this so called exterior calculus and use the fact that tensors in oblique coordinate systems under (affine) transformations produce same calculus as the invariant theory of (differential) forms [67, Section 9].…”

Section: E N D N O T Ementioning

confidence: 99%

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Energy based methods applied in mechanics by using the extended Noether's formalism

Abali

2023

Z Angew Math Mech

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Physical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether's formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether's formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.

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Section: E N D N O T Ementioning

confidence: 99%

“…22Often, its study is conducted by using differential forms [66]. We will not make much use of this so called exterior calculus and use the fact that tensors in oblique coordinate systems under (affine) transformations produce same calculus as the invariant theory of (differential) forms [67, Section 9].…”

mentioning

confidence: 99%

Energy based methods applied in mechanics by using the extended Noether's formalism

Abali

2023

Z Angew Math Mech

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“…We would like to stress out the constitutive laws are strictly local. That is, they are maps between cotangent spaces, 18,19 that is, between fibers over a point of a manifold, and the Hodge map over a point has no effect on the fibers over its neighboring points.…”

Section: Hodge Operator and The Constitutive Lawsmentioning

confidence: 99%

Systematic derivation of partial differential equations for second order boundary value problems

Kettunen

Rossi

2022

Int J Numerical Modelling

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Software systems designed to solve second order boundary value problems are typically restricted to hardwired lists of partial differential equations. In order to come up with more flexible systems, we introduce a systematic approach to find partial differential equations that result in eligible boundary value problems. This enables one to construct and combine one's own partial differential equations instead of choosing those from a pre-given list. This expands significantly end users possibilities to employ boundary value problems in modeling.To introduce the main ideas we employ differential geometry to examine the mathematical structure involved in second order boundary value problems and exploit electromagnetism as a working example. This provides us with an organized view on the key building blocks behind boundary value problems.Thereafter the approach is naturally generalized to a class of second order boundary value problems that covers field theories from statics to wave problems. As a result, we obtain a systematic framework to construct partial differential equations and to test whether they form eligible boundary value problems.

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“…They obey the anti-commutation relation { γ a , γ b } = 2η ab , where η ab is the usual Minkowski metric. To study the Dirac field in curved spacetimes, we need to introduce the vierbein field [44]. This field consists of an orthonormal set of four vector fields that serve as a local reference frame of the tangent Lorentzian manifold at each point of spacetime such that…”

Section: Entanglement In Dirac Fields In a 1+1 Flrw Universementioning

confidence: 99%

Cosmological quantum entanglement

Martín-Martínez

Menicucci

2012

Class. Quantum Grav.

131

179

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Abstract. We review recent literature on the connection between quantum entanglement and cosmology, with an emphasis on the context of expanding universes. We discuss recent theoretical results reporting on the production of entanglement in quantum fields due to the expansion of the underlying spacetime. We explore how these results are affected by the statistics of the field (bosonic or fermionic), the type of expansion (de Sitter or asymptotically stationary), and the coupling to spacetime curvature (conformal or minimal). We then consider the extraction of entanglement from a quantum field by coupling to local detectors and how this procedure can be used to distinguish curvature from heating by their entanglement signature. We review the role played by quantum fluctuations in the early universe in nucleating the formation of galaxies and other cosmic structures through their conversion into classical density anisotropies during and after inflation. We report on current literature attempting to account for this transition in a rigorous way and discuss the importance of entanglement and decoherence in this process. We conclude with some prospects for further theoretical and experimental research in this area. These include extensions of current theoretical efforts, possible future observational pursuits, and experimental analogues that emulate these cosmic effects in a laboratory setting.Cosmological quantum entanglement 2

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Differential Forms, with Applications to the Physical Sciences.

Cited by 82 publications

References 0 publications

Energy based methods applied in mechanics by using the extended Noether's formalism

Energy based methods applied in mechanics by using the extended Noether's formalism

Systematic derivation of partial differential equations for second order boundary value problems

Cosmological quantum entanglement

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