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The Shaggy Steed of Physics
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.
Fair warning: the review is lengthy, because the book demands it.
Yes, I believe on the inside cover the book EULA stating "All reviews of this material must be over 3 pages in length"
Do really dense people warp space more than others?
...is O'Reilly's Physics for Game Developers.
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.
The Army reading list
Maybe I'm in a minority of 1, but that review didn't seem very long to me. Sure its longer than a jacket summery... but it hardly does as far enough to be in-depth let alone deserve a warning.
anyway... better than the usual 'contents table' affair we get on slashdot I suppose. Hardly Sunday paper review long though.
0daymeme.com: Great stuff.
Cheers,
Erick
http://www.busyweather.com/
Huzzah! Interest has been loosed upon the internet by phyl0x, the great emancipator. No longer shall interest be bound and forced to live and work as a slave to...well...ummm....
/petpeeve
Oh, I see... It looks like you meant to use the word "losing," as in "lose, losing, lost." Good luck with that next time.
Otherwise I can't recommend it -- the book is gibberish if you can't follow the math.
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.
Kind of inappropriate if you ask me.
you obviously haven't seen "A Brief History of Time", "In Search of Schrodinger's Cat", "Schrodinger's Kittens" and many other non-maths physics books.
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".
1,082,811
The only acceptable defense of scientific results is to say that they were the product of the Scientific Method.
The don't explain why things fall, but they explain with superb beauty and conciseness how things fall. Asking why things happen is verging into the realm of philosophy, rather than physics.
MIT Press blurb
The book is also online in html form. It sounds like you weren't used to the Lagrangian formulation of mechanics, which has been around for a long time but is usuually not taught in lower level undergrad physics courses (i.e. normal engineering physics). If you take an upper level class in classical mechanics, you'd cover it thoroughly. Sussman and Wisdom's book presents it in an interesting computer-inspired way. Note though that this is a textbook (with problem sets and all that), not a popularization.
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.
Not quite; the restricted three-body problem, where one of the masses is infinitessimal compared to the other two, can be solved analytically. The solutions reveal the existence of five points where the net effective force on the massless third body vanishes -- these points being, of course, the Lagrange points familar to students of orbital mechanics.
I'm surprised that the reviewer found so much of the material new; do college physics courses these days not include classical mechanics and the like?
Tubal-Cain smokes the white owl.
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.
Ok, a little layman summary:
There's a fairly easy problem in physics. It's called the two-body problem. In it, you model (or predict) the motion of two objects in space as dictated by the force of gravity.
It's based on the Newtonian equation for gravity, which is that the force of gravity acting on two objects is proportional to the square of their distances. To put this more simply, the force of gravity between two objects gets drastically weaker as they are moved farther away.
All that being said, the main thrust of the book is apparently related to the three-or-more body problem. In it, the same basic equation is used. But since every body is being influenced by every other body, which are in turn being influenced by every other body, it gets very messy. Well-nigh uncalculateable, at least by people. The calculus just becomes too complex.
Fortunately, the two-body problem establishes a good enough model, allowing for us to model the motion of planets in our solar system, so long as we take into account that there's some wobble we have to throw in.
Now, I know this didn't explicitly cover the math, but basically, the book takes all of what I just said and builds it up from very basic to very complex mathematics.
Or thats my understanding, anyway.
It's not what you know, or even who you know- It's how many people recognize your damn
Gravitons (the gravitational force carrying particles) are still very much hypothetical. They are postulated merely because all other forces seem to have a particle that carries them. Certain quantum theories of gravity require them, but no one has a really good quantum theory of gravity yet anyway. However, the fact that gravity does not act instantaneously has been observed. So there does need to be some way to propogate gravity from one location to another. There does need to be some type of wave, particle, or both transmitting gravitational information at the speed of light (or very close to that speed).
This book sounds pretty cool, but I disagree with the reviewer regarding the level of the book, which I can gauge from the reviewer's comments. The reviewer tends to think it's well beyond advanced undergraduate physics classes, but from the material involved I think it's somewhere between the intro and advanced undergrad classes. It sounds like this book would be useful for armchair physicists that would like to get their hands a little more dirty, people minoring in physics, and physics majors wanting a little more 'oomph' before their 'real' classes kick in. But IMHO, one definitely shouldn't need to be a grad student in math or physics to enjoy this book as the reviewer implies.
For example, the reviewer writes "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."
So upon reading that one assumes the reviewer at least took some decently advanced calculus-based physics classes well beyond the freshman level (like a two-semester class of E&M or quantum mechanics, or classical mechanics).
But then the reviewer says "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."
This quote right here reveals that the reviewer hasn't been exposed to any 'advanced' physics classes, maybe just advanced introductory ones. Only the intro classes will 'tell' you about Kepler's 2nd law and conservation of angular momentum. This concept, though, is usually proved and derived from the fundamentals in any reasonable undergraduate physics mechanics class beyond the freshman-level class. Such an undergraduate level mechanics class would, for example, use the textbooks by Arya or Marion/Thornton.
Similarly with motion in phase space, simple harmonic motion, Lagrangian equations of motion, the energy eigenstates of the hydrogen atom (this would be in the quantum mechanics class), etc. These are all topics which are examined from the fundamentals, and encountered usually within the first two or three years of an undergraduate physics curriculum.
So the Shaggy Steed is a book somewhere beyond the intro physics classes, but not as difficult as the more advanced undergraduate physics classes, where the majors start going. Note - if you really like this low-level sort of stuff, though, you might seriously consider majoring or minoring in physics.
So I disagree when the poster writes "It's a beautiful book if you're a graduate-level student of math or physics..." Most of the material covered seems to be the standard fare that the typical undergraduate physics major will encounter, and some of these topics will likely be encountered several times prior to graduation.
make world, not war
"I recommend reading some Wittgenstein for more insights and CLARITY[?!]"
yes, and when you're done with that, round it all off with Kant, Hegel, and Sartre to make everything just peachy, crystal, transparent.
or should I be merciful and shoot you first?
The prize-winning N-body book referred to in the parent is Leslie Greengard's 1987 PhD thesis, "The Rapid Evaluation of Potential Fields in Particle Systems".
-Tom Duff
That your average engineer, chemist or other science-minded college-educated person is not at least comfortable with Lagrangian mechanics is a failure of physics education.
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.