Customer Reviews

New Foundations for Classical Mechanics (Fundamental Theories of Physics)

5 star:		(4)
4 star:		(2)
3 star:		(3)
2 star:		(0)
1 star:		(0)

 

 

David Hestenes is a forerunner of the modern development of Clifford algebra. His current research activities can be followed in the site http://modelingnts.la.asu.edu/GC_R&D.html. 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 http://www.mrao.cam.ac.uk/~clifford/.

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.

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.

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.

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.)

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...

This is surely one of the best introductory books to start from, learning the Geometric Algebra (Clifford Algebra). The author is generous in exposing his phlosophical views, especially on the meaning of number concept and how to extend it. This is surely very valuable, though nobody can get %100 percent of what another one is trying to convey in written form. Ofcourse this also, and perhaps mostly, depends on the conformity between the ways of thinking and orientations of the two brains: Those of the author and reader. In this respect I am fully indebted to him in that I have gained new points of view. For example the multiplication of two real numbers, and especially that of a real number (scalar) and a vector, is no more a pre-given external operation to the Geometric Algebra since both the scalars and vectors are members of this same set and thus their multiplication is nothing different than that of other elements of this algebra. May be not for some body else, but for me, this is a new and profound approach. Also many thanks are due to the historical insights and highlighting, again quite generously, provided in the text.

Yet the axiomatic development of the subjext given Chapter 1 is unfortunately rather sloppy. But it should be obvious that axiomatization, by definition, if not carried out rigorously may become quite confusing, if not annoying. This should be the same, whether the treatment is introductory or not. For example, yet just at the start of the axiomatization process, the algabre being defined is expressed to consist of the sums of all k-blades (Eq.7.1). Whether this is a loose definition or the result of something following is by no means clear. Only upon very carefully studying and striving, it is understood that the a.m expression, although very loosely introducing the terms of k-blade and k-vector, is not a definition. It might well be an insightful theorem which could be given a recursive-constructive proof. But alas! No explanation and not a single word on this. Being axiomatic is, obviously, not the equivalent of dryness but rather of rigour. Thus it should not, by no means, exclude presenting explanation where danger of confusion is clear. The same confused style of presentation unfortunatly pervades and continues untill the end of this chapter, where the axiomatic foundations are laid. What should actually be recursive definitions of k-vectors are given as axioms in Eq. 7.13a and 13b. Without the slightest explanation on the meaning of equality of two members in this algebra, the reader is put face to face with an unnecessary puzzle in exercise 7.1, which the author qualifies as some logical fine point.

Any body looking for some rigour surely shall need a companion text, of which the best, as far as I know, is "Clifford Algebra to Geometric Calculus" which the same author co-authors. Finally and for the sake of fairness, I should make it explicit that this book I am reviewing is quite better than the much praised "Geometric Algebra for Physicists " by Chris Doran-Anthony Lasenby in respect of all above criticisms.

This is the perfect example of what I humbly believe every mathematics textbook should aspire to be like. Written in a highly motivational style, and with problems that are difficult but just hard enough to make sure you really "got" the material, it is by far one of the most enjoyable and rewarding books I've ever worked through.

There are very few, if any, instances common in other texts where a problem is set without giving the reader enough previous grounding, making it an ideal self-study book (which is what I used it for). If you can't do a problem, then you can be sure you didn't absorb the previous material well enough.

And if you don't know just how cool geometric algebra is when you start, you're in for a series of startling surprises!

Despite its name, I've seen no "new foundations" at all. Perhaps "new methods" would be better, but are they actually new? I learnt many concepts given in this book from a German book on engineering published at the beginning of the 20th century (including the "magical" i). The notation has been improved (dramatically, indeed), but I can understand why this formalism failed to succeed long ago, since most of the explanations in this book can be done more easily and intuitively with vectors and dyads--even Hestenes uses "tradicional" vectorial tools like cross product and pseudo-vectors very often. He is fond as well of topics we can find in old books on Mechanics, like ballistics and hodographs. Still, this is a valuable book for those who want to learn geometric algebra or to see Mechanics from a different point of view, even if, IMO, it is essentially a notational and computational trick.

Update: After re-reading it I would lower the rating to two stars: Lagrangian formalism is elementary and minimal, there is nothing on continuum mechanics, and it has some mistakes.

The entirely new approach to the mathematical treatment of familiar Physics situations. A very useful tool for a Physisist.