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.
A lighter physics book... (Score:5, Interesting)
One of the chapters - on 'real world' projectile motion - is available for download at the above site, so you can get a feel for the writing and content.
Re:Huh? (Score:1, Interesting)
The Shaggy Steed of Physics For Idiots (Score:3, Interesting)
Cheers,
Erick
If you want it to make sense... (Score:5, Interesting)
If you want it to make sense, you gotta accept the fact that the book, by itself, is not supposed to turn an interested laymen into a learned professor. Books like these, for me, spur me to go learn the basics instead. Even if I never get all the way through the book, I can at least use it to tell me what I need to know to be considered "learned" in the field.
I remember in college as a CS student, being spoon-fed the easy-to-learn computing theory and feeling like I was getting nowhere. I picked up the Hopcroft & Ullman automata book and was, at the time, completely inundated by the math (I went to a commuter college with a not-so-advanced math & CS dept.). But at least I knew what I really needed to learn next. I ignored the professor pretty much for the rest of the class (and never opened the textbook) and instead investigated only those things I required to understand the H&U book. I found that by the end of the class, though I was not yet a quarter of the way through the book, I knew a lot more than my classmates, who still struggled with the basic concepts of the field.
If the book seems too much for anyone other than an grad student, try using it instead as an index of things you need to learn first. Don't know those formulas? Look 'em up. Even if you don't grasp everything in your target book, you'll be smarter for it in the end.
Re:Most of them (Score:3, Interesting)
Too light (Score:5, Interesting)
Basic problem with building a game physics engine: if you do all the obvious stuff, it sort of works. If you're competent, you should be to that point in a few months. Getting from "sort of works" to "works" is about 5x to 10x as hard as the first step. There are really only a few game physics engines out there that really work.
You'll find out more about stiff systems of nonlinear differential equations than you ever wanted to know, if you don't give up first.
It's interesting that the book talks about the problems that occur when you take into account the propagation delay of gravity. Game physics engines, having rather large time steps, have some similar problems. I'll have to read this and see if I get any new insights applicable to game engines.
There's a related book, an ACM prizewinner, on the N-body problem. There's a clever numerical solution to the N-body problem that works for large N (millions), so you can simulate galaxies forming and such. The basic idea is that you can treat a group of bodies as a single body if they're near to each other and far away from the body being affected. This can be quantified and safe limits computed for grouping. It's thus a numerical solution with a proveable upper bound on the error, which bound can be made arbitrarily small at the cost of more computation. This is effectively as good as a closed-form solution, although some older mathematicians deride it as inelegant.
Re:The restricted three-body problem... (Score:3, Interesting)
Not quite; your example is not a three-body problem, but really a two-body problem in disguise. The equations of motion for the two finite masses can be solved separately, since they are not influenced by the infinitesimal mass. Then the problem reduces to a single particle (the one with infinitesimal mass) travelling in a time-varying field.
Re:The Shaggy Steed of Physics For Idiots (Score:3, Interesting)
The book's content seems to be the basics of classical mechanics (harmonic oscillation, Kepler's laws), and some other stuff like hydrogen atom in quantum mechanics, special and general relativity, etc. As the review indicates, it's primarily things involving the 2-body problem. You may have learned this stuff at a higher level in other physics or chemistry classes, but this book will give a sense of where these concepts come from, and how it's not just fancy professors with crazy beards plucking them from thin air.
You can read my other comments on this topic, but I disagree with the poster as to the level of this book. I place it somewhere beyond the introductory physics classes, but not as difficult as the advanced undergraduate physics classes a physics major would take. The reviewer, on the other hand, implies one needs to be a graduate student in math or physics to enjoy this book, but I disagree with that. [disclaimer - i AM a graduate physics student, however]