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New Foundations for Classical Mechanics (Fundamental Theories of Physics) [Paperback]

David Hestenes , D. Hestenes
5.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

30 Sep 1999 0792355148 978-0792355144 2
This book provides an introduction to geometric algebra as a unified language for physics and mathematics. It contains extensive applications to classical mechanics in a textbook format suitable for courses at an intermediate level. The text is supported by more than 200 diagrams to help develop geometrical and physical intuition. Besides covering the standard material for a course on the mechanics of particles and rigid bodies, the book introduces new, coordinate-free methods for rotational dynamics and orbital mechanics, developing these subjects to a level well beyond that of other textbooks. These methods have been widely applied in recent years to biomechanics and robotics, to computer vision and geometric design, to orbital mechanics in government and industrial space programs, as well as to other branches of physics. The book applies them to the major perturbations in the solar system, including the planetary perturbations of Mercury's perihelion.
Geometric algebra integrates conventional vector algebra (along with its established notations) into a system with all the advantages of quaternions and spinors. Thus, it increases the power of the mathematical language of classical mechanics while bringing it closer to the language of quantum mechanics. This book systematically develops purely mathematical applications of geometric algebra useful in physics, including extensive applications to linear algebra and transformation groups. It contains sufficient material for a course on mathematical topics alone.
The second edition has been expanded by nearly a hundred pages on relativistic mechanics. The treatment is unique in its exclusive use of geometric algebra and in its detailed treatment of spacetime maps, collisions, motion in uniform fields and relativistic precession. It conforms with Einstein's view that the Special Theory of Relativity is the culmination of developments in classical mechanics.

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Product details

  • Paperback: 724 pages
  • Publisher: Springer; 2 edition (30 Sep 1999)
  • Language: English
  • ISBN-10: 0792355148
  • ISBN-13: 978-0792355144
  • Product Dimensions: 23.7 x 15.6 x 3.8 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 243,472 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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About the Author

David Hesteness is awarded the Oersted Medal for 2002.
The Oersted Award recognizes notable contributions to the teaching of physics. It is the most prestigious award conferred by the American Association of Physics Teachers.

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5.0 out of 5 stars Fascinating! 4 Jun 2013
Format:Paperback|Verified Purchase
Hestenes had dedicated most of his career to developing Geometric algebra which, by some aberration, the maths community glossed over. They seemed to decide that Clifford's geometric algebra meant algebra rather than geometry. As a consequence the physicists have been left to fend for themselves and produced some beautiful routes to deeper intuitive understanding and easier practical means of working in multidimensional spaces. This book (I'm told) is the most accessible entry into the subject. I certainly find it very good as far as I've gone. Not least for the many excellent diagrams and exercises based on actual application of the ideas rather than bald examples of doing theoretical proofs, which so many algebra books use as a cop-out, leaving the student with working tools and no idea of where to apply them other than following like sheep. The very practical outlook of the book in using real geometrical insight and showing the profound benefit that the geometric product gives over mere exterior algebra in both solving and penetrating the exploitation of graded structures for separating out scalars and invariant features is masterly. The student aware of these insights is much better equipped to go on into differential geometry, complex analysis and eg. projective or conformal geometry or invariant theory. It's certainly helped me considerably in the latter topics.

The main disappointment about the book is however that geometric calculus, the development into differential geometry, is not covered in the same volume, though instead the many examples across a wide range of practical applications give the physicist many valuable insights into many areas of mechanics..
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Amazon.com: 4.1 out of 5 stars  9 reviews
37 of 38 people found the following review helpful
4.0 out of 5 stars A problem with relativistic mechanics... 1 Dec 2005
By Prof C. R. PAIVA - Published on Amazon.com
Format:Paperback
David Hestenes is a forerunner of the modern development of Clifford algebra. His current research activities can be followed in the site [...] Probably his most important book until now (written with Garret Sobczyk) was "Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics" (Dordrecht: Kluwer Academic Publishers, 1984) also available at Amazon.com. This book on the new foundations for classical mechanics (second edition) was written as an introduction to geometric algebra. The term "geometric algebra" was coined to stress that this formulation of Clifford algebra is a unified language for physics and mathematics; it is not a matrix algebra (as used in quantum mechanics in the disguised forms of Pauli and Dirac matrices) as it uses a new property, the contraction, which makes it different from other associative algebras. A recent book on geometric algebra is "Geometric Algebra for Physicists" by Chris Doran and Anthony Lasenby (Cambridge: Cambridge University Press, 2003) - see the site [...]

Geometric algebra is a graded algebra based on the geometric product of vectors which reduces to the inner product (a scalar) when the two vectors are parallel and to the outer product (a bivector) when the two vectors are orthogonal. The geometric product is associative and can be used in spaces with any dimension (as opposed to the cross product of vectors which is not associative and can only be used in three or seven dimensions). Therefore, the geometric product is able to generate several graded algebras: (i) in two dimensions we recover the complex numbers as elements of a real algebra, not as elements of a field; (ii) in three dimensions we get a geometric algebra that is far better than the Gibbsian approach mainly due to the geometric role of rotors is reflections and rotations; (iii) in four dimensions we obtain the so-called spacetime algebra which is perfect for Minkowski spacetime within the context of special relativity - see the paper from Hestenes in American Journal of Physics (vol. 71, pp. 691-714, June 2003). Hamilton's quaternions are properly understood. Even as a new gauge theory of gravity on flat spacetime Hestenes' geometric algebra plays a very important role - see the paper from Hestenes in Foundations of Physics (vol. 25, pp. 903-970, June 2005). The clear and insightful approach that geometric algebra can bring to the Dirac equation is also remarkable.

My only problem with this book is due to Chapter 9 on relativistic mechanics. In this chapter Hestenes takes the usual approach that can be found in traditional four-vectors, by representing an event as a paravector, i.e., as a sum of a scalar and a three-dimensional vector (in Euclidean space). This kind of approach doesn't take advantage of geometric algebra (as in his article on spacetime algebra for Am. J. Phys.) because spatial vectors are not directly linked to an observer (and to its proper time) as they are in spacetime algebra where the so-called space-time split clearly leads to an invariant and proper formulation of physics. In Chapter 9, indeed, these paravectors induce a relativistic approach and not a proper approach. Nevertheless, apart from this remark, my overall comment on this book is very positive.
32 of 35 people found the following review helpful
5.0 out of 5 stars Great for physicists, okay for others 12 Sep 2000
By Thouis Jones - Published on Amazon.com
Format:Hardcover
This is a great introduction to the Geometric (Clifford) Algebra. It's fundamentally a physics textbook, however. Those readers whose only desire is to learn the Geometric Algebra might feel some frustration at having to separate out the Geometric Algebra from the physics. Readers that prefer learning by exploring applications and examples will like this book; those that prefer explanations in the abstract will still enjoy many sections, but will have to make it through the more applied sections to get the full story.
Reading the book and working through the problems gives a firm grounding in the use of the Geometric Algebra and teaches classical mechanics besides. I could easily recommend this book as a physics textbook on its merits in that area alone.
25 of 30 people found the following review helpful
3.0 out of 5 stars Doesn't stand on its own 13 Jun 2005
By C. Andersen - Published on Amazon.com
Format:Paperback|Verified Purchase
While I found this a reasonably well-written text, I judge a book based upon how well it stands on its own - that is, can I read the book, work through the exercises, and acquire a grasp of the topic. While this is a much clearer and more accessible intro to geometric algebra than Hestene's "Clifford Algebra to Geometric Calculus", it is more the type of book which would accompany a class in GA, where the instructor fills-in the gaps, rather than a stand-alone text. The worked exercises are relatively few, and are typically of the nature: start with this, magic, more magic, resulting answer. It lacks sufficient explanation, is not self-contained, but this can be partially overcome with additional resources.
6 of 6 people found the following review helpful
4.0 out of 5 stars Can Geometric Algebra be Taught in High School? 27 Jun 2010
By Pdecordoba - Published on Amazon.com
Format:Paperback|Verified Purchase
Update, 9 December 2012: Don't miss Alan Macdonald's Linear and Geometric Algebra, which is recommended enthusiastically by Hestenes.

Although New Foundations for Classical Mechanics (NFCM) is primarily a physics book, it's also intended to demonstrate the usefulness of geometric algebra (GA) in solving any sort of problem whose data and unknowns can be formulated as vectors.

Several previous reviewers were more qualified than I to discuss the advanced aspects of this book. I review it from the viewpoint of someone who was considering Hestenes' advice, expressed elsewhere, to employ geometric algebra in high-school classes. Of course I didn't expect that New Foundations would be suitable for high schoolers. Instead, I wanted to decide whether GA might save students enough time in college to be worth introducing in high school. To that end, I worked many of the problems in the first 3-1/2 chapters, then skipped to chapter 5, where I have worked on only the first section. I also attempted, with mixed results, to solve classic geometry problems via GA, especially those involving construction of circles tangent to other objects.

That amount of experience is probably necessary to decide about trying GA in high schools. My own decision is a cautious "yes", with some caveats regarding both GA itself, and this book.

First, NFCM is definitely not a stand-alone textbook. Although Hestenes' explanations of many topics are not only lucid, but genuinely thought-provoking, few people who tackle NFCM on their own will find it easy. But then, Hestenes never said it would be. As he noted on p. 39 of his Oersted Medal paper (see first comment, below, for all references in this review),

"... I had to design [New Foundations] as a multipurpose book, including a general introduction to GA and material of interest to researchers, as well as problem sets for students. It is not what I would have written to be a mechanics textbook alone. Most students need judicious guidance by the instructor to get through it."

By the way, anyone who's considering teaching GA anywhere should read that paper to learn from Hestenes' own travails.

Since I had no instructor to give me judicious guidance, I read several papers on GA by Hestenes and others. The lectures and problem sets from Cambridge University were helpful up to the point where they became too advanced for me. Another good reference was Ramon Gonález Calvet's "Treatise of Plane Geometry through Geometric Algebra". The chapters from the previous edition of NFCM that Hestenes maintains online offered many valuable perspectives.

However, all of those resources couldn't make up for the lack of a good solutions manual, with plenty of additional worked-out examples. If I could make just one suggestion to Hestenes for facilitating adoption of GA, this would be it. Ideally, the manual would also show how to explore GA using computer software such as GAViewer, or even CaRMetal (which I plunked along with). I suspect Hestenes would agree with all of these recommendations.

IN SUMMARY
This is a good book for learning to use GA, if used as Hestenes intended. I'm convinced that GA is worth trying to teach at the high school level. I don't expect that it would be any easier to teach than the geometry and trig that it would replace, but it should pay off better down the road.

Please note that Hestenes and his colleagues have also done extensive research on teaching physics. The "Modeling Instruction in Physics" method they developed has given good results. (See links.)
15 of 19 people found the following review helpful
5.0 out of 5 stars Excellent place to start learning Clifford Algebra. 6 Aug 2000
By Theodore Erler - Published on Amazon.com
Format:Hardcover
A briliantly pedagogical introduction to Clifford Algebra as a unified algebraic language for Newtonian Mechanics in three dimensions. The book is full of applications and nonstandard approaches which simply cannot be found anywhere else. This is essential reading for anyone interested in Clifford Algebras or who wants a deeper appreciation for classical mechanics. This is a lot of book...
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