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.
Current Amazon sales rank. . . (Score:3, Informative)
1,082,811
Another very good book (Score:5, Informative)
MIT Press blurb [mit.edu]
The book is also online in html form [mit.edu]. 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.
The restricted three-body problem... (Score:5, Informative)
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 [wolfram.com] 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?
Layman's translation (Score:3, Informative)
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.
Re:Small nit-pick (Score:3, Informative)
Re:Slightly OT.. (Score:3, Informative)
They do have a math markup, mathml.
It is not real nice to use without some sort of editor to generate it. I think MathType does it in Windows.
I think the latest Mozilla supports it:
http://www.mozilla.org/projects/mathml/
htt
Usually, it is probably better to make a pdf, but then you miss out on hyperlinks (unless you know how to stick them in your pdf)
Re:Most of them (Score:5, Informative)
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.
Related reference (Score:5, Informative)
Sounds like a more advanced mechanics text (Score:3, Informative)
In my first year in grad school, I took a great classical mechanics course taught by a guy who uses classical mechanics in his research on planetary systems. His name is Stanton Peale. He got semi-famous by publishing a paper just before Voyager arrived near Jupiter, saying that Io might be volcanic. He would have published it a lot sooner, but he didn't notice that orbital data on the Galilean moons are, for historical reasons, recorded differently than those for other moons in the solar system. He had therefore mistakenly calculated that none of the Galileans would be volcanic. By chance (if such a thing exists
But I digress... in Peale's class, we used the standard graduate text on Classical Mechanics, which is Goldstein's Classical Mechanics [amazon.com].
Both the Goldstein book and the Marion & Thornton book cover Lagrangian and Hamiltonian mechanics. Goldstein goes into more details about things like Poisson Brackets and canonical transformations.
The Landau & Lifshitz book Mechanics [amazon.com], the first volume of the "Course of Theoretical Physics," covers much of the same material, but is quite concise. For that reason, like most of the Landau/Lifshitz (and Lifshitz/Pitaevskii, after Landau died) books, it is pretty dense.
I'm not sure if Oliver intended to bring these things to folks other than physics majors, but who other than physics majors (and maybe the occasional math major or other science/engineering major) has enough interest in the subject to wade through the math? The math isn't all that complicated (for a physics or math major), but it's complicated enough to deter anyone not really interested in the subject. Peale's classical mechanics class was not quite a weed-out course, but it was one that a significant number of people dropped in their first year and were taking for the second time when I took it. I worked really hard in that class and ended up learning a lot. And it wasn't just the math that made it tough. But the point is that this material can be taught at a level that's challenging for grad students...
--Mark
Re:Math Explains Nothing (Score:3, Informative)
Eigenstates of said *operator*. And the operator does not "reduce a wavefunction to an eigenstate" -- a *measurement* does (allegedly).
Re:Slightly OT.. (Score:3, Informative)