The moon, of course, is not a fixed distance for the earth. This is most readily apparent in autumn (northern hemisphere); closest approach is typically 75 to 90 minutes before moonrise, so you won't normally be able to see it, but you will be able to see the moon get smaller as it comes up over the horizon and rises into the sky.
But if you could see closest approach, under the right conditions, the effect can be quite spectacular. This image, taken with a special camera that shows light not normally visible to the human eye, shows the impact of the moon against the ozone layer.
If you look carefully, you'll see bands of shading showing the waves radiating outward from the impact - much like you would see dropping a stone into a puddle. The effect isn't quite the same (the moon doesn't actually drop through, puddles aren't round, and ozone (O3) being inherently symmetric, has a characteristic flow that differs from the lopsided H2O molecule.
If you have a head for the math, Phil Plait describes this in even more detail over at Bad Astronomy.
Work of late has required quite a bit of time playing with SVG images, and consequently decrypting the descriptions of Bézier curves.
This in turn got me to thinking about approximating probability distributions over bounded intervals. If you don't know the true distribution, you may be able to make reasonable predictions by using a simplified distribution that shares the same characteristics.
http://fliptomato.wordpress.com/2006/12/30/from-griffiths-to-peskin-a-lit-review-for-beginners/
Is the Landau "Theoretical Minimum" published anywhere? (by preference, in English)
The project is wrapping up. To keep my brains from going into my feet, I've been reviewing Aitchison & Hey.
I stared at problem 4.1 until I was stupid. Or maybe I started that way. Does anybody get that far into relativistic QM without having it burned into their souls that taking the complex conjugate of an expression reverses the order of the operators?
In other words, if we start with:
AΨ = 0
then we know that another way we can spell this is
Ψ †AΨ =0
Further more, we can take the transpose of the first equation:
(AΨ)† =0
and multiply on the right
(AΨ)†Ψ =0
Now all the fiddly bits do what they are supposed to, and you don't find yourself staring at things like Ψ V Ψ† wondering how you are supposed to know if that commutes or not. You take the difference, and the real bits fall out, and the imaginary factors do sensible things, and the main difference between the two is that the operators act on other things, and right answers just start raining down from the sky.
At least, that's what happened when I tried it. 'course, the text gave me a pretty clear understanding of what answer I was supposed to get; it is possible that my progression from the problem to the solution is completely lacking in local logical continuity.
I watched Raiders of the Lost Ark during the holiday (I refuse to buy the box set for one movie, so took advantage of an opportunity to borrow a copy).
In the scene where Dr. Jones is explaining Tanis to Army Intelligence, there are physics problems on the blackboard. It appears to be a trajectory problem of some sort - the leading term is the height displacement of a projectile given its initial vertical velocity, but there's some sort of correction. Can anybody identify it?
Instead of getting sleep last night, I spent a bunch of hours working on Hermite Polynomials. Yeah, yeah - I could have looked everything up on MathWorld; that's not the point.
[more ]If you are associated with MIT, IAP 2006 includes he Feynman Films, sponsored by Markos Hankin.
I'm not associated with MIT, but that's OK... because the Education Development Center cleared its backlog, and sent me the remaining pieces of the Cornell Lectures I had been waiting for.
Oddly enough, the package appeared the same day that I called to ask when I might expect the order to be completed.
(Minor quibble: technically speaking, I don't quite have the kit, because I haven't asked the nice people at Tuva Trader to send me a copy of "Take the World from Another Point of View". Fine, I'll watch the Robb lectures or the Dirac lecture instead. That works for me.)
"The law that entropy always increases - the Second Law of Thermodynamics - holds, I think, the supreme position among the laws of physics. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations - then so much the worse for Maxwell's equations. If it is found to be contradicted by observation - well, these experimentalists do bungle things from time to time. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation." — Sir Arthur Eddington
Lifted from Wikipedia, and really too long for my quotes page, so here it is.
The Archimedes Death Ray. Pirates should avoid MIT on sunny days.
I discovered some trig identities this week; as they don't look even a little bit familiar, I assume they didn't come up when I was paying attention in highschool.
In short: for any product consisting of sines and cosines: if there are an odd number of sines, the expression can be rewritten as a sum of sines. If there are an even number of sines, the expression can be rewritten as a sum of cosines.
You can get a quick sense of how this might kind of work out by considering the exponential expression for sign, which has a factor i in the denominator. An odd number of sines in the product produces an odd number of i factors, producing an imaginary number. To get real numbers from this, you'll need to pop one of the i's off again by creating sine terms.
It's all pretty straight forward to do with exponentials and a little bit of induction; or you can demonstrate it using the more common identities....
[more ]Working on yet another physics exercise (more on that later, I hope), I found myself needing the trigonometric formulas for half angles.
Demonstration below the fold....
[more ]Whee, not only did the nice people at allmediashop.com send me (via Amazon) the Feynman audio lectures that I wanted, but the folks at The Tuva Trader sent me the Feynman video lectures that I wanted.
Work has bothered me for a while - the idea that the change in energy could be related to the force times a distance seems really weird to me. So I did some number crunching to figure out what was going on.
[more ]Demonstrating the randomness of my corner of the universe, I twigged to working on the scalar and vector potentials recently. I'm not sure how much of my library covers them.
Feynman scatters his material across several chapters of Volume II
II-2 Differential Calculus of Vector Fields
II-15 Vector Potential
II-25 Electrodynamics in Relativistic Notation
Carver Mead also discusses it in Collective Electro Dynamics, and there is some information in Griffiths.
But I wasn't quite finding what I was looking for - a demonstration of the theorems from which the scalar and vector potential follow.
[more ]Introducing the Gauss Rifle. I particularly like the explanation, which strikes me as quite simple and elegant. Which is probably indicative of how much time I've spent pouring over non-relativistic QM problems of late.
Via Joel on Software, some articles I have to look into....
"A probable prime is an integer that satisfies a condition also satisfied by all prime numbers."
And I'm only three years late in learning that Agrawal had a break through leading to a polynomial time prime test.
Oh how very cool - a transcript of teaching base two arithmetic to a class room of third graders (which I suspect means age 9, for the most part, given that the class was taught in May).
I really like the way the contributions of the audience are presented in the transcript. It really communicates to me the idea of a classroom of kids getting excited by the discoveries they were making.
I was a lot more impressed on the first reading, though, when I had missed the age range - I had initially thought the audience was much younger.
Hat tip: Greg Morrow.
Of late, I have come to the conclusion that I never really learned integration by parts when I was studying calculus. Or perhaps that I learned it by rote, rather than by understanding. So I recently took steps to correct that....
Great pictures of a droplet striking a flat surface, which are shown to vary with the pressure of the gas in which the experiment is conducted.
Robert Vanderbei provides a Gravity simulator, which illustrates a number of different solutions to the multibody problem.
Most, of course, are unstable - meaning that any deviation from a perfect orbit causes the next orbit to get worse. The program illustrates this, as the numerical analysis used has only finite precision, and the error will accumulate.
But Ducati3 (and my personal favorite Ducati3_10) are stable - the potential pulls small errors back toward the idea orbit. Very hypnotic.
If you are more into blowing stuff up, look at LagrangeN or LagrangeNa.
Vanderbei was kind enough to forward a link to an additional paper, New Orbits for the n-Body problem, that I wasn't able to find when navigating the site.
Pekka Parviainen on mirages in Finland. Many good photos of inferior and superior mirages caused by thermal layers above the sea.
Quantum Chromodynamics Made Easy.
See also quark gluon plasma, explained by Dr. Elmo.
I was given a copy of Gribbin's Feynman biography for my birthday, and am finishing the final chapters.
Late in the book, I discover that it references a path integral solution to the hydrogen atom. Wait a moment, says I, as I dash to the bookshelf - sure enough, I already have Kleinert waiting for me.
I wonder if a crack habit would be cheaper. I can quit any time, rilly.
Stenger writes "Any vector operator J whose components above the angular momentum commutation rules [Jx,Jy] = iħJz will have the following eigenvalue equations: J^2|j,m> = j(j+1)ħ^2|j,m> where J^2 = Jx^2 + Jy^2 + Jz^2."
Landau and Lifshitz volume 3 had a demonstration of this that I could follow.
[more ]As part of a lecture on the relationship between symmetry and conservation laws (Volume I, Chapter 52), Feynman casually drops that the symmetry under a shift in quantum mechanical phase is related to the conservation of electrical charge. Huh?!?!?
Victor Stenger wrote a very good introductory level paper on gauge symmetry. At least, I think it's introductory level, but I've been pouring over a lot of this stuff over the past few months, so maybe it isn't all that easy.
Anyway, the paper explains Feynman's comments well.
[Updated: the paper has been pulled, but appears to be available elsewhere. Link updated 2004.07.17]
Newtonian physics suffered two damaging blows in the 1900s - it broke down where the world was small and where the world was fast.
I've been focusing my attention on the former. Here's some interesting visuals of the latter, which help address the question what does the world look like travelling at relativistic speeds, by animating a world where c is of the order 5m/s.
If you could pitch a fastball at the speed of light, it would still look round to the hitter. That's really messed up.
Now, when I come across something like a Runge Kutta formulation, it tends to go into my notebook. Once upon a time, I was capable of doing math, and I'm trying to reacquire the skill. I don't yet feel the need to work it out for myself, as much as go through the derivation exercise once, to force myself to understand it.
In other words, I hit the web, looking for an explanation.
Searching reveals a number of explanations of the general approach, several derivations of 2nd order RK (because that's the easy one to derive), and discussion of 4th order RK (because that's the useful one, and the formula is pretty, if you are into that sort of thing). But almost nothing on 3rd order, and nothing to help somebody like me who gets the idea, but can't get the symbols to come out right.
Once I finish this, there will be.
[more ]I've found, in various learning environments, that I do best when things go wrong. I get a much fuller understanding of an exercise when my first answer requires debugging.
So I suppose the typos in the text should be accepted as a blessing, since they increase the probability of a debugging session.
Matching typos; these I regard as blatantly unfair. After all, one of the checkpoints is whether an equation is consistant with its neighbors. Eventually, I gained enough confidence that the book was wrong to check it against the original paper, which by good fortune was near to hand.
I think this indicates I've gone beyond the reasonable point with my recent book purchases. And three more are supposed to be on the way.
Learn about Manning's equation and other topics in open channel hydraulics.
Yes, I run into this sort of thing in casual conversation. Sean spent the weekend playing Christopher Robin, and watching the rivers turn around. So on Monday we started working out the math.
Forget Fear. Worry about the addiction. Notes on aerial physics courtesy of Trapeze School New York.
Snoopy Dance!
I've finally had a breakthrough working on Feynman's wobbling plate problem.
[more ]A beautiful image by Kevin Pease, for his comic Why the Long Face?.
Of course, as I've been doing way to much Feynman of late, I notice that the colors of the secondary rainbow are in the wrong order. Also, the sky between the bows should be darker, and I think we should be seeing sunset coloring - the bow seems awfully high for the sun to still be giving full day light.
M.G.J. Minnaert's Light and Color in the Outdoors has a lot of math in it explaining why these things look the way they do. Don't pick up this book if some other task will shortly be requiring your attention.
With F.S. Woods started on its journey back to the library, the physics bookshelf now looks like
[more ]It seems I was in error. I was browsing yesterday, and discovered that Feynman's Least Action lecture has been published on cassette 6 of Volumne 19: Masers and Light.
Also, David Morin of Harvard has published a chapter on The Lagrangian Method that I found helpful for getting a better grasp on how to pick the right Lagrangian.
The latest arrival - Quantum Mechanics and Path Integrals.
Happy happy happy.
The table of contents. Thanks IndiAndy
Of course, Volume II Chapter 19 is missing..., and I don't have a cassette player in the car.
One of my fans passed along a copy of Advanced Calculus, by Frederick Shenstone Woods.
I'll have to return it, but for the moment I'm quite happily occupied.
In the lost lecture, Feynman relates various eccentric points of the velocity circle to different conic sections. This led me to explore parabolas, and the reflection properties of them.
[more ]I listened again to the lecture while driving this evening, and managed to catch the quiet hint Feynman dropped for another geometric demonstration of the reflection property.
It's a bit hand wavvy right now, so I'm not too happy with it. Of course, we've already got a geometric demonstration, so we know it's right. I'm doubt I'll worry further about it, now that the basic idea is off my chest.
[more ]The recording of Feynman's Lost Lecture does not end when the bell rings; the tape is still rolling when the physicist answers some questions afterwards. Last night, what caught my attention is his comment that there's another way to think about the reflection property of the ellipse.
If you consider the tacks and string construction, as you move about the ellipse, string is being shed on one side and taken up on the other. These have to balance out, because the length of the string is fixed.
I haven't gotten anywhere yet trying to apply this idea geometrically, but analytically it works quite well.
[more ]Dauger Research presents a few very cool simulations of hydrogen atoms.
With less motion, but more explanation, see The Orbitron
Feynman-Bob says check it out.
Images of Earth and Jupiter, as seen from Mars.
Update: It seems the original photos looked more like these
I've been working my way through Feynman's Lost Lecture (planetary motion). I'm not quite finished with it, but along the way I've picked up on a number of points that weren't immediately clear to me from the audio portion and the written companion.
[more ]
