Clifford Algebra to Geometric Calculus

Hestenes, David; Sobczyk, Garret

doi:10.1007/978-94-009-6292-7

Cited by 828 publications

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“…We represent spacetime by a four-dimensional vector manifold, as defined in Hestenes and Sobczyk (1984). Let x = x(x 0 , x 1 , x 2 , x 3 ) be a spacetime point parametrized by coordinates x µ , where µ = 0, 1, 2, 3.…”

Section: Framesmentioning

confidence: 99%

“…According to Hestenes and Sobczyk (1984), the fundamental differential operator on a vector manifold is the derivative ∂ = ∂ x with respect to a point x on the manifold. All other differential operators can be expressed as algebraic functions of this operator.…”

Section: Derivativesmentioning

confidence: 99%

“…Although it is best to define ∂ independently of coordinates, as done in Hestenes and Sobczyk (1984), if the manifold (or some part of it) is parametrized by coordinates, the point derivative ∂ can be obtained from the coordinate derivatives by…”

Section: Derivativesmentioning

confidence: 99%

“…It was originally introduced (Hestenes, 1966) as a unified mathematical language for physics, with applications to electrodynamics, quantum mechanics, and gravitation. It has since been generalized and developed into a comprehensive geometric calculus (GC) (Hestenes and Sobczyk, 1984) with a wide range of computational capabilities. This paper shows how computational problems in gravitation theory can be simplified with GC.…”

Section: Introductionmentioning

confidence: 99%

“…The present computational method is a straightforward application of the general method of fiducial frames developed in Chapter 6 of Hestenes and Sobczyk (1984). I will employ their results without repeating the derivations.…”

Section: Introductionmentioning

confidence: 99%

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Curvature calculations with spacetime algebra

Hestenes

1986

Int J Theor Phys

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Section: Framesmentioning

confidence: 99%

Section: Derivativesmentioning

confidence: 99%

Section: Derivativesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Curvature calculations with spacetime algebra

Hestenes

1986

Int J Theor Phys

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Simo-Vu Quoc rods using Clifford algebra

McRobie

Lasenby

1999

Int. J. Numer. Meth. Engng.

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SUMMARYWe present an alternative derivation of Simo and Vu Quoc's numerical algorithm 1 for modelling the non-linear dynamic behaviour of rods. The original derivation uses di erential topology, describing large rotations using the Lie group SO(3) and Lie algebra so(3), but resorting to quaternions for the numerical implementation. The new derivation uses Cli ord or geometric algebra as developed by Hestenes 2; 3 for both formulation and implementation. We contend that the new approach is considerably simpler to follow, and thereby allows alternative modelling strategies to be more readily investigated. The new description is also novel in that all formulae for rotational kinematics are applicable in a Euclidean space of any dimension. Copyright ? 1999 John Wiley & Sons, Ltd.KEY WORDS: Cli ord algebra; geometric algebra; rods; large rotations BACKGROUNDOver the past few decades the foundations of classical mechanics have been rigorously reformulated in terms of di erential topology. The major contributors to this process have been Abraham and Marsden in America, with their deÿnitive tome 'Foundations of Classical Mechanics' 4 and Arnol'd in Russia, whose work in this area is typiÿed by his book 'Mathematical Methods of Classical Mechanics'. 5 The early contributions focused on classical particle mechanics and rigid body mechanics. A subsequent advance was the publication of Marsden and Hughes 'Mathematical Foundations of Elasticity' 6 where the di erential topology reformulation was extended to elastic continuum mechanics, providing a fundamental alternative to the earlier description based upon classical tensor analysis as exempliÿed by the works of Truesdell, Noll, Gurtin, Naghdi and others in Vols. III, IV and VI of 'Handbuch Der Physik'. 7 This reformulation has continued apace into the realms of computational continuum mechanics most notably through the work of Simo and coworkers, in particular, on the ÿnite element analysis of the large displacement behaviour of rods and shells (e.g. References 1,[8][9][10][11]. This presents a particular problem in that di erential topology rarely forms part of an engineering education, and thus the foundations of applied mechanics are now written in a language that is impenetrable to most engineers. * Correspondence to: F. A. McRobie, Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 lPZ, U.K. E-mail: fam@eng.cam.ac.uk Over the last ten years an alternative reformulation has been pioneered by Hestenes and co-workers who, building upon Cli ord algebra and Grassman calculus, have developed a mathematical framework they refer to as Geometric Algebra. In the introductory 'New Foundations of Classical Mechanics' 2 and in the more formal 'Cli ord Algebra to Geometric Calculus' 3 the foundations of this alternative approach are presented. Again any physical emphasis is on particle mechanics. The initial extension of the approach to continuum mechanics has been commenced by Gull et al. 12 In this paper we provide the initial extensions to co...

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Clifford algebra, symmetries, and vectors

Altmann

1996

Int. J. Quantum Chem.

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rnThe realization of a Clifford algebra in laboratory space is considered and it is demonstrated that the elements of the algebra cannot, as often assumed, be directly identified with vectors in this space, but, rather, that they form the parametric space of the symmetry operations of the Euclidean group as performed in the laboratory space. Details of this parametrization are established and expressions are given that determine the action of the Euclidean-group operations (screws included) on laboratory-space vectors in terms of the elements of the Clifford algebra. A discussion of Clifford vectors, bivectors, and pseudoscalars and their relation to the Gibbs vectors is provided. The correct definition of axial and polar vectors within the Clifford algebra is carefully discussed. It is shown how simple it is to generate finite point groups in 4-dimensional space by means of the Clifford algebra.

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Clifford Algebra to Geometric Calculus

Cited by 828 publications

References 0 publications

Curvature calculations with spacetime algebra

Curvature calculations with spacetime algebra

Simo-Vu Quoc rods using Clifford algebra

Clifford algebra, symmetries, and vectors

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