<i>New Foundations for Classical Mechanics</i>

Hestenes, David; Vold, Terje G.

doi:10.1119/1.16386

Cited by 249 publications

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“…The first is that the geometric algebra of spacetime is the best available mathematical tool for theoretical physics, classical or quantum. (4)(5)(6) Related to this part of the program is the claim that complex numbers arising in physical applications usually have a natural geometric interpretation that is hidden in conventional formulations. (5"7-9) David's second major idea is that the Dirac theory of the electron contains important geometric information, (1'3'1°'m which is disguised in conventional matrix-based approaches.…”

Section: Introductionmentioning

confidence: 99%

Imaginary numbers are not real—The geometric algebra of spacetime

Gull

Lasenby

Doran³

1993

Found Phys

110

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This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a "geometric product" of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytie functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes" spacetime algebra, which automatically incorporates the geometric structure of spaeetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained--results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.

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Section: Introductionmentioning

confidence: 99%

Imaginary numbers are not real—The geometric algebra of spacetime

Gull

Lasenby

Doran³

1993

Found Phys

110

View full text Add to dashboard Cite

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“…Moreover, it is important for establishing relations to other mathematical structures. There are good reasons, for example, for basing the theory of Lie algebras on the commutator product in Geometric Algebra (see [8] and remarks in [9]). …”

Section: The Commutator Productmentioning

confidence: 99%

“…Although our central concern here is with projective geometry, it is worth noting that, for the case of Euclidean signature, numerous applications of the foregoing relations among three points are given in [9] using the language of geometric algebra. Note, for example, that the relation of A to b and c in (4.2) is exactly that of the conventional vector cross-product.…”

Section: Algebra Of Points and Linesmentioning

confidence: 99%

“…It represents a single resultant force only if it degenerates to a 2-blade. In general, a wrench represents a resultant force and couple (see Section 6.1 of [9]). …”

Section: Line Geometrymentioning

confidence: 99%

“…Detailed applications of geometric calculus have been worked out for a large portion of mathematics [12] and nearly the whole of physics *This work was partially supported by NSF grant MSM-8645151. [7][8][9]. It seems safe to claim that no single alternative system has such a broad range of applications.…”

Section: Introductionmentioning

confidence: 99%

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Projective geometry with Clifford algebra

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Distance Geometry: Theory, Algorithms, and Chemical Applications

Havel

1998

Encyclopedia of Computational Chemistry

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RMSD D root-mean-square coordinate difference.2 DISTANCE GEOMETRY: THEORY, ALGORITHMS, AND CHEMICAL APPLICATIONS which is just the symmetric part of their geometric product ab. Thus it is natural to define the outer product of two vectors as the antisymmetric part of their geometric product, i.e., a ^b D 1 2 ab ba 4 Ã 2 D e 1 e 2 e 3 2 D e 1 e 2 e 3 e 3 e 2 e 1 D e 1 e 2 e 2 3 e 2 e 1 D e 1 e 2 2 e 1 D e 2 1 D 1 8 a ^ b ^c D a ^b ^c D a ^b ^c 10 a ž b ^c D 1 2 a b ^c b ^c a D a ž b c a ž c b 11(and similar identities involving Ãe 3 e 1 and Ãe 2 e 3 ), we find thatMore generally, multiplication by the unit pseudo-scalar Ã maps vectors (bivectors) to their duals, which lie in the planes (lines) perpendicular to them, and have the same magnitudes.Let us now consider Gibbs' triple product, which is a scalar equal to the signed volume of the parallelopiped spanned by the three vectors. Using the fact that pseudo-scalars commute with everything in the algebra: a ž b ð c D 1 2 aÃ b ^c C Ã b ^c a D Ã 4 abc acb C bca cba D Ã a ^b ^c

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New Foundations for Classical Mechanics

Cited by 249 publications

References 0 publications

Imaginary numbers are not real—The geometric algebra of spacetime

Imaginary numbers are not real—The geometric algebra of spacetime

Projective geometry with Clifford algebra

Distance Geometry: Theory, Algorithms, and Chemical Applications

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