The Shaggy Steed of Physics 181
| The Shaggy Steed of Physics: Mathematical Beauty in the Physical World | |
| author | David Oliver |
| pages | 300 |
| publisher | Springer |
| rating | 8 of 10 (if you have the required math skills) |
| reviewer | Sarusa |
| ISBN | 0387403078 |
| summary | Beautiful but demanding examination of the two-body problem. |
The force on each body, whether gravitational or electric, is proportional to the square of the distance between the bodies. An isolated sun and planet form such a system, and a hydrogen atom, which is just a proton and electron, can be simplistically modeled as such. This may seem a trivial problem: you can sum it up in half a page in a physics book. But that's because all the detail work has been done for you. Furthermore, anything more complex than the two-body problem is chaotic and incapable of exact solution, so it's up to the two-body problem to carry us along. This is a complex problem, so this review is rather lengthy.
Let me warn you right off the bat that this is not a book for the faint of heart. It kicked my ass. The concepts are fast and furious, and the math is dense. Equations festoon the pages, daring you to ignore them. But you may not, they're fundamental to the discussion. Mr. Oliver opines that anyone with basic undergraduate math should be able to handle it. I had calculus, differential equations, and a good dose of physics in college and I still found the book tough going, mostly due to the whirlwind of notation and sheer number of variables introduced. I ended up keeping a cheat sheet of key definitions which ended up being four pages long, and took almost two weeks to process it. It reads like an advanced college physics book, except without extra examples or redundant explanation -- he expects you to be smart or motivated enough to keep up.
As an example: 'Using Hamilton's equations to eliminate p' and q', the total rate of change may be compactly expressed as df/dt = df/dt + [f,H] where [f,g] is the Poisson bracket of any two functions of the motion: [f,g] = (df/dqi*dg/dpi - dg/dqi * df/dpi)' I've reformatted this slightly for text limitations; he of course doesn't use * for multiplication, and you should read all 'i's as subscript i. This is fairly simple math in the context of the book.
So now that I've scared you off, what's the payoff? Well, unlike my college physics books which just lead me from factoid to factoid there are moments where the hard work pays off in big "oooh" moments. Your book might give you Kepler's second law: a planet sweeps out equal areas of its ellipse in equal times. But why? We'll just call it 'conservation of angular momentum'; that should hold you plebes. But in Shaggy Steed you'll find the equations like this that you might have thought were fundamental falling out of the woodwork, built up from the real fundamentals.
We start out by defining coordinate spaces and deciding that we're interested in Newtonian/Galilean rather than Einsteinian physics for the moment, since our subjects travel slowly enough and relativity makes things nastier. We start with a particle that has two vectors -- position and velocity. Turn this into two ensembles of rigid body particles exerting force upon each other. From this we build up the laws of motion, arriving at the total energy H of the system, and the 'gene of motion,' the Lagrangian: the difference between the kinetic and potential energy. 'Gene of motion' is a pretty bold claim, so we are shown how every mechanical quantity of the system may be derived from the Lagrangian. From there it's on to the 'action' principle, which is basically the integral of the Lagrangian over time - the key being that of any path the particles may take, they act in a way to minimize the action. Every other law of motion (including Newton's) follows from this, though to explain why it's the case we need general relativity. This was my first 'oooh' moment.
Chapter 3 really sets the pace for the rest of the book. If you're thrown off here, you're not going to make it out alive. To summarize: "Motion consists of the trajectory flow of particles in phase space. Each isolating invariant introduces a degeneracy into the motion in which the full phase space available to the trajectories degenerates into a submanifold. Increasing numbers of isolating invariants correspond to increasing degeneracies of the motion which restrict the trajectories to increasingly restricted submanifolds of phase space." This is more or less the programme of the entire book. Dig out as much complexity as required, then simplify to solvability.
Oliver introduces each new concept, so if you're following along carefully, you can follow along. This is all done half in equations, so we're diving so deep into math that you (okay, I) may be several pages in and forget where you were coming from and where you were going. Then suddenly you're out the back end and he nails it all with a beautiful concrete application or insight. For Chapter 3 it's Hooke motion, which you can think of as approximating two weights connected by a spring. Now if you've ever taken differential equations, or dynamics, you're probably uncomfortably familiar with this system. Now here it is all laid out for you, everything explained, and boy those resultant equations look mighty familiar. So that's where that all comes from, and why they use those particular symbols. The linear central force and the inverse-square forces of our two-body problem turn out to be closely related as well.
To be crushingly brief, Chapter 4 finally gets down to the (relatively) practical matter of classical planetary (Keplerian) mechanics, and why four dimensional spheres are special. Chapter 5 dives into quantum mechanics, and the hydrogen atom loosely simulated as a two body problem, since it has only the nucleus and one electron. And let's derive the fundamentals of quantum physics and the periodic table while we're here. Though I've neglected to mention it till now, Oliver doesn't neglect the human side of all this. He doesn't linger on it, but he does provide context. It's amusing to see how many of these inexorable equations were originally derived by geniuses like P. Dirac, only to be disowned because the implications were too outlandish.
In Chapter 6, it's time to step out of Newtonian/Galilean space and into Einsteinian space. We've made a lot of assumptions, such as the infinitely fast propagation of forces. This is no longer the case; time is no longer separate from space. In fact, we learn how to rotate space into time through imaginary rotation angles (known as 'boosts'). e=mc^2 falls out. But our shaggy steed eventually breaks down on the precession of Mercury. In the land of general relativity, even a simple two-body problem is really a many-body problem - forces are no longer instantaneous, they require force particles. The steed is of no more use.
But wait! Chapter 7, The Manifold Universe, takes on many-body motion like Don Quixote tilting bravely at a windmill, and tries to pull some order from the chaos. KAM theory is introduced and our many-body problem turns out to be not absolutely chaotic, but a mixture of regular and chaotic motion. You may have noticed that our many-body solar system doesn't just fly apart. We can model it more or less as a set of two-body problems with minor perturbations (minor being the key). And of course we can model fluids even though the internal motion is chaotic. Order emerges. Our shaggy steed is revived, transformed.
The back of the book contains the Notes, which are compact digressions into the hard (yes ...) math. I have to admit some of them completely lost me. But they're not required, just extra reading for those of you who eat this stuff up.
This all leaves me with a bit of a quandary. It's a beautiful book if you're a graduate-level student of math or physics, smarter than me (your best bet), or willing to put a lot of effort into it. Otherwise I can't recommend it -- the book is gibberish if you can't follow the math. I can't help but think that it would make a fantastic course in the hands of a skilled practical math teacher like Dr. Gary Sherman at RHIT; I certainly could have used his help with this. So, it's to teachers like him that I'd really suggest this book, for eventual dissemination to their students. Or if you dig physics and have the math skills, you might want to try riding "The Shaggy Steed of Physics" alone. If it throws you, there's no shame.
You can purchase The Shaggy Steed of Physics: Mathematical Beauty in the Physical World from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
Long review? (Score:4, Insightful)
anyway... better than the usual 'contents table' affair we get on slashdot I suppose. Hardly Sunday paper review long though.
shaggy lost dog story (Score:3, Insightful)
Re:The Shaggy Steed of Physics For Idiots (Score:4, Insightful)
Re:Most of them (Score:4, Insightful)
These are "physics books" the way "the matrix" is a computing and AI primer. That is to say, they tell you that several of the important concepts exist, in a way that's entertaining, but don't do much to tell you how to actually _use_ them.
At best, "physics overview for the layman", as opposed to "physics reference".
Re:Math Explains Nothing (Score:4, Insightful)
Slightly OT.. (Score:3, Insightful)
Feel free to mod me as such, but the review reminded me how horribly mathematics is represented in a browser. Wouldn't it be great if one day we could simply type:
Re:Math Explains Nothing (Score:3, Insightful)
Here's a clue: a model of the Universe (or parts of it) is just that - a model. Meaning it describes the howaccurately enough, but does not explain the why . And guess what - causality is part of the reason.
But hey, don't let reason stand in the way of a good troll. This is
Re:Math Explains Nothing (Score:3, Insightful)
Math indeed does not explain anything, but it also does not require any physical explanation. Mathematical propositions are true or false in their own realm which is entirely distinct from the physical realm. I recommend reading some Wittgenstein for more insights and clarity on this matter.
Re:Math Explains Nothing (Score:2, Insightful)
Without math, you can learn a little something qualitatively of modern physics by reading some of the popular physics books of the day, like Brief History of Time, etc. Many of these authors convey nicely at a high level how and why things happen.
But if you want to know the details, you need math. Quantum mechanics is interesting because it's like a manifestation of linear algebra. Why does an operator reduce a wavefunction to one of the eigenstates of said wavefunction? That concept is one of the most central concepts to quantum mechanics, yet you wouldn't understand what eigenstates or wavefunctions are without some knowledge of math. If you explain it using only words, you're still beating around the bush, and basically it's the math that you would be describing.
Finally, to prove your example is crap, please explain using words why things fall. If your description involves anything about gravitons or Higgs bosons, please explain why they form, why gravitons should be spin-2, what a spin-2 boson field implies, etc, without using math. Basic answer - you cannot.
this is another failure of physics education (Score:5, Insightful)
In physics we generally don't think in terms of Newtons Laws, but rather in terms of the action and fields.
In my view, the lower level college physics classes which teach 18th century physics are a complete waste of time (as the review points out, all those laws fall out of more fundamental principals). The engineering students who are forced to take physics are not even given the chance to learn "real" physics, and the physics and other science majors who take it will simply be told to forget it and learn a better way of thinking a year later.
I'm always asking people in my department (I'm a physics grad student) why in the world we teach these useless classes. Generally the defense is that people wouldn't learn the concepts if we taught them the real way, that the math would be too hard, and people would get caught up in it.
They forget what it was like as an undergrad. Physics can be hard, even old, 18th century physics. When I've taught physics, people always get caught up in the math. The best we can do is to at least teach the right way, and introduce the right concepts. The math can be taught, packaged or explained.
There has been very little effort that I have seen to put real physics concepts in a package which is understandable by your average freshman biology student. This book is obviously no exception. It does not have to be this hard, and physics does not have to be only for physicists. Why do we insist on complicated terminology and crazy sounding descriptions?
I know that a lot of engineers and others out there have had more modern classical physics classes. Were they any good? Was your education in physics enlightening or frusterating? These issues really bug me, and I hope some of you out there have had better than I've seen.
Why this book is signifigant (Score:2, Insightful)
Why you ask?
Well, I've got a pretty good math background and I've read some (not all) of the Feynman lectures. So while the math of advanced physics doesn't scare me (okay, it scares me a little), I lack any physical intuition.
I wasn't quite prepared to plow through a dry 500 page book on mechanics. However, I was looking for an entertaining read.
The reason is that mechanics is the intuition of physics. Most mathematicians can run mathematical circles around their physicist counterparts. Ever ask a physicist to "prove" something? However, ask a mathematican to "calculate" anything complicated in physics and you'll usually stop them cold (with a few notable exceptions [Von Nuemann, Kolmogorov, etc]).
Mechanics is the practice of doing physics calculations. If axioms and proofs are the tools of mathematicians, then fundamental laws and calculations are the tools of physicists. All introductory physics books (including Feynman) dance around the calculations of physics. Sure, when you've finished reading the Feynman lectures, you can pontificate on basic E&M, QM, etc. You'll be able to describe all kinds of interesting phenonmena. You just can't calcuate anything.
I've haven't read this book, but I think I will. If it is as entertaining as the reviewer says it is, then I could imagine it might become quite the classic.
Of course, this is just the opinion of a stupid math major....
Re:this is another failure of physics education (Score:3, Insightful)
I study in Goettingen (I'm sure you know where that is.) and the mathematics department has a large hall of models of surfaces of least action, mainly done between 1900 and 1950. After the advent of computing most of the problems they were working on were done computationally.
I would just comment further that Newtonian mechanics has the enormous advantage of working mainly in directly measurable values, making errors easier to identify. In fact if you go into engineering, there's almost no system of calculation which doesn't work in directly measureable values, and so I think it simply makes sense to teach simple physics in those terms too.