The Principle of Least Time, originally due to Fermat, is such a principle, but it is incomplete in that the more general formulation requires that the optical path length must be stationary, meaning that there can be situations where light takes an extremal path other than a minimum.  That calculus talk is all fine and dandy provided you have a good numerical feel for what it means to analyze a stationary point.  Feynman encourages us to take such a stand…

We can easily use brute force in the m-file Least Time to illustrate that the least time and least distance paths vary significantly.   The least time path has the smallest optical path length, which is the product of the distance traversed by light times the index of refraction experienced during the traversal.  We can further plot the optical path length (opl) vs. the distance along the mirror axis and carry out some numerical analysis that Feynman alludes to.

First, we take the same setup from FLoP: Reflection and Least Distance, and this time work with opl’s instead of distances by simply taking our distance formula results and multiplying by n = 1.5 in water.  Points A and B are again randomly assigned, and point c calculated based on those.  Here’s the result:

The crooked path has a shorter time of flight according to the numbers.  We see that the ratio of indices of refraction are related to the simple geometry by Snell’s Law, giving us a check on our brute force calculation.

In referencing Fig 26-5, Feynman has us picture a function that has a minimum, zoom in on that minimum, and see that nearby points have a negligible first derivative but small second derivative.

Recall that a first derivative is just the slope of the tangent line at the point in question, so that if it is nearly zero then its tangent line is horizontal and thus unchanging. Points surrounding a minimum are literally dead points, their pulses flat-lined.  The second derivative indicates if the original function is concave up (has a min) or concave down (has a max).  A positive second derivative is for concave up since as we move farther to the right away from the minimum, the function is increasing.  It’s as if we take those surrounding dead points, compares their pulses (slopes), and depending on which one is closer to zero then the rest, tells you whether the function is overall increasing or decreasing or both on either side, in the case of an inflection.  To make a plot for this, we first need to numerically compute the derivatives.  You basically picture a square grid to serve as your coordinate system, where each grid mark is traversed left and right by dx and up and down by dy.  A derivative is thus dy/dx, but you must indicate which grid point you’re at, so it’s more like [y(i+1)-y(i)]/eps for a uniform spacing eps on x.  If you then step up the y(i+1) to y(i+2) and the y(i) to y(i+1),  take the difference and divide each by eps, then subtract those and again divide by eps, you arrive at the second derivative.  There are multiple variations on this process, as you can instead opt to step half-way on either side of a grid point, which leads to a more accurate numerical derivative.  But we can choose a small enough eps such that this way is plenty accurate.  We thus have the following MATLAB function to compute the 1st and 2nd variations

function [dx,ddx] = variation(y,i,eps)
dx =(y(i+1)-y(i))/eps;
ddx =(y(i+2)-2*y(i+1)+y(i))/eps/eps;

(To run this function with the m-file Stationary to produce the following graph, save the function in a separate m-file titled ‘variation’ and run it in the same workspace as the m-file.)

Point c is the red mark corresponding to the least time path, whereas the black mark is the straight line (shortest distance) path.  The first derivative at c is zero to four decimal places, while it is nearly -0.3 at the black mark, indicating that that the straight line path is not near a minimum and its slope is decreasing.  I picked an eps value so small that the red and black marks are separated by about 15,000 ticks in the index i, meaning if I type in variation(opl, i+100, eps) into the command window, I will still be computing a 1st derivative close enough to point c to give me a value close to zero.  I get .002, actually.  Notice that for both marks the 2nd derivative is substantial and positive, indicating a significant numerical variation past a 1st derivative and an overall concave up function.

From a sheerly mathematical viewpoint, we see that a derivative at one point tells you a lot of information about function on a larger scale.  In fact, if we know all of the derivatives at a single point, we can reproduce the entire function via a Taylor expansion.  That’s just amazing!  If you’re working with derivatives, you’re necessarily studying dynamics, which is what physics is all about.  So we can place a concrete example to the abstract mathematical notions and create some plots to see that the individual numbers are indeed behaving in this fashion.  Something about that is immensely satisfying.

Feynman stresses that the physicist needs to assign physical meaning to the equations, whereas the mathematician can just keep hammering away with a rigorous approach to everything.  He in essence says that the two attack a given problem in completely different ways, albeit work with the same mathematical machinery.  When researching something new, the physicist need not bother with organizing his thoughts in a coherent framework that can then methodically be applied to gradually get to the heart of the problem.  Rather, he let’s his intuition do that, and relies on conservation laws and simple models to try to get a grip on the problem, letting it simmer in his head for weeks or months on end, until finally one day, insight gleams forth under some relaxing state of mind, and everything falls together.  The mathematician, meanwhile, works hard to put everything in place before she ever begins.  She likes her problems well-posed, for then she can reach into her bag organized case of tools and bring her training to bear on the problem in a straightforward manner, deriving the result, establishing convergence, proving uniqueness, etc.  That process certainly requires ingenuity and sometimes strokes of genius, but it follows one after the other in a laid out procedure.

The physicist follows a different line of attack.  His mind is quite cluttered.  I would compare him to a curious craftsman trying to open up a black box, unaware of what tool he might need next.  He doesn’t have the luxury of an organized workspace; he must keep all of his tools at his fingertips.  Sure, he possesses a deep understanding of mathematics.  Otherwise, it’s unlikely he could ever forge connections among similar concepts in physics, let alone among the vast fields of physics, that are crucial to making significant progress.  But he doesn’t concern himself with the likes of real analysis past a certain stage of utility.

I sort of like my black box analogy because it captures, if only in a highly simplified and over-generalized way, the interplay of these two fields.  The boxes are the problems from physics, and the tools to open them are mathematics.  History has shown that after opening one black box, the physicist is presented with another inside, as if he had just opened a Russian nested doll.  Excited, he gets to work on trying to open the next black box, while the mathematician often occupies herself with polishing up the one he just opened.  Or she might be developing a tool which he needs to open the next box, yet neither has any clue that such is the case at that point.   Occasionally the tools to open the box have been around for many years but the physicist has not learned to use them.  And certainly many a mathematician take a stab at opening the boxes themselves, more often than not only to pry it open part way until one day a physicist comes along and shakes it up and down, pours out all of its contents, and leaves it swinging from its hinges.

It used to be that mathematicians took all of their problems from physicists for the most part.  However, now that math is infiltrating other fields at an overwhelming pace, mathematicians are beginning to learn less and less about physics.  Yet the reverse is not true, while the concepts of physics are also finding numerous applications in other areas.  It remains to be seen what will come of all of this, but I can say that being in a relationship with a mathematician has only heightened my respect for them overall, as the skill set they develop truly is unique.

A typical setting for digesting thought-provoking material is not the front seat of a Pontiac in the midnight to early morning hours at below freezing temperatures.  It was the last week of December in 2003, and the country was on Orange Alert, which had prompted a massive deployment of security guards at spots thought to be at high-risk for a terrorist attack all throughout the US.  I had signed up for a graveyard shift guarding one of the city’s main water towers to make some quick cash.   Aside from turning my engine on for a five-minute perimeter-check every hour, I was to stay parked in a dirt lot overlooking the tower for my entire eight-hour shift, making sure I radio in every 20 minutes to let them know I am still awake.  My friends called this a most unbearable job description.  I thought it was better than a minimum duty work-study job at a library.

Clutching a flashlight in order to read Feynman’s words in the blackness around me, it slipped my mind that I was still cold with three layers of clothing on top of thermal underwear.  Captivated by his coherent account of the field I hoped to dedicate my life to, I took in his sense of wonder and admiration for our ability to really understand.  I would periodically glance around in bemusement at the phenomena around me, such as the subtle green glow emanating from the LED of my walkie talkie, which was neatly spread about on the frozen precipitation of my windshield.  I was looking out into the night sky pondering General Relativity, confident that with Feynman guiding me through the language of nature, I could grasp its deepest secrets.  It was an enlightening experience, a crossroad between boyhood wonder and sophisticated thinking, one that I look back on when I am in need of comfort for pursuing (as a first choice) a career with dim prospects for employment.

Every field has its heros.  The nice thing about making Feynman one is that he is not nearly as mythical in the popular press as Einstein, yet he was certainly a cut above the rest.  I agree with Julianne’s take on Feynman, though I had no qualms about worshiping him as an undergrad.  Feynman worship in graduate school should be minimized, as that is setting yourself up for failure (or for revitalizing Wal Street).  However, his lectures should continue to be utilized, as they are brilliant.

Feynman’s description of the problem poses a method of brute force attack.  First, his excerpt:

It is easy enough on a computer to construct the set-up in Fig 26-3 on a coordinate system.  We construct the 1st and 4th quadrants as running from 0 to 1 on the x-axis and 1 to -1 on the y-axis.  Point A is placed at some height on the y-axis and point B at some negative height on x=1.  We can then use the distance formula to compute all the possible paths from A to B as the light ray hits every point between D and F.  The smallest distance corresponds to point C.  The m-file prompts for how many divisions to make from D to F and computes both the analytic solution based on y=mx+b and checks it with the brute force method of comparing each distance.  The result is plotted below:

As I am reviewing basics physics by reading through some chapters in the Feynman Lectures,  I often feel compelled to take this one step further.  Feynman was a brilliant theorist who could have easily confounded readers by lecturing at a mathematically over-sophisticated level, but rather he describes his subject in a very applied, intuitive manner.  His lectures are strewn thick with what I call “numerical nuggets”, indicating his ability to perform numerical analysis in his head, if only at qualitative level.  You can tell that he arrives at an analytic expression using calculus only after he deeply ponders the limiting process underlying it.  To fully benefit from Feynman’s insight, therefore, I am attempting to quantify some examples of this.

Since FLoP is an acronym for the Feynman Lectures on Physics with an appropriate numerical pun, it makes for a good title, I think.  And it certainly describes my blogging effort overall as viewed by my girlfriend, who deems it an utter failure.  In any case, it provides me additional motivation to not review college physics passively and at the same time keep up with my coding skills.

I am currently reading chapter 26, Optics: The Principle of Least Time.  As an easy example, consider Feynman’s description of refraction studies prior to the discovery of Snell’s Law

We can take a few angles in air and verify the interpolation Feynman mentions in parenthesis.  The fact that they fit to a parabola is indicative of the nonlinearity of Snell’s law.  The following code will give us a plot:

air = [10 30 60 80];
water = [7.5 22 40.5 48];
plot(air, water)
title('Refracted angle vs. incident angle at an air-water interface')


We can then go to tools, basic fitting, select ‘linear’ and ‘parabolic’ and check ‘show equation’, to produce

from which we see that the linear fit is way off.

For the most part, time constraints will prohibit me from walking through the MATLAB code, so I’ll ordinarily toss the m-files in a paste-bin and just present the results.

End of Week 1 Group Photo

I attended the program last year and would summarize it as the single most memorable experience pertaining to physics that I have ever had.  The group of students in attendance were top-caliber, and I made friends with almost all of them, some of whom I still keep in touch with on facebook.  Lots of them were already involved with GWA in one way or another, a couple having internships lined up at LIGO and a couple more doing research for their advisers on data analysis or numerical relativity.

On The Wave, the island's bus system

One in particular (far right, on the right) persuaded me into a year long furlough away from computation, having enticed me into experiment with his descriptions of what a joyous time he was having researching the optical system that will be placed in Advanced LIGO.   (He was a great conversationalist and spoke five languages, in fact–one of them Russian, which came in handy when I met an attractive Russian girl on the bus to class; he scored me a date with her.)  Combined with the enthusiasm Dr. Saulson expressed in his lectures, my newfound interest in optical science took on a life of its own and I ended up deserting gravitational waves within a couple months, took an experimental physics class the following semester, and applied to the College of Optical Sciences at the University of Arizona to do a PhD in optics last December.  With the rejection later came the realization that I am not fit for the laboratory, but my interest in optics has persisted.  Luckily, there’s plenty of computation to be done in that field.

My memory is fading, but I believe the schedule for the summer school was 9am to 4pm with an hour for lunch between morning and afternoon sessions and two fifteen minute coffee-breaks within each session.  The layout was very conducive to helping vacationers serious students digest as much material as possible.  There would be a lecture followed by some sort of problem solving session (except in Dr. Saulson’s case, as he preferred to give two lectures…fine by us, since that meant no homework.)  The first week covered Peter’s talks on the experimental aspects of GWA in the morning and Dr. Creighton‘s lecture and homework session on the relevant astrophysics of the gravitational wave sources in the afternoon. The second week dove into numerical relativity in the mornings by Dr. Gonzalez, followed by the pertinent data analysis techniques in the afternoon, presented by another Peter, Dr. Peter Shawhan.

Engaged with MATLAB and Dr. Shawhan's Exercises

I enjoyed Dr. Shawhan’s section the most, as he prepared a fantastic series of exercises to work on after his lecture, which was done in the adjacent room with computers seen above.  The exercises covered the basic methods to extract a gravitational wave signal from the raw stream of data collected, which is full of all sorts of noise.  This was to be performed using–you guessed it–MATLAB!  I contemplated re-solving all of the problems and posting about them, and finally decided that would be way too much work.   But then I noticed that Peter posted solutions this year!  If physics is your thing, then studying his solutions is by far the most rewarding way to go about learning MATLAB.  I hope to document in more detail the code for my favorite problems and maybe add a bit to them in a post, as only a one-liner is needed to play the sound of an incoming gravity wave signal (reminiscent of my last post, Hacking the Pay Phone).

Week 1 had a good deal of homework for the astrophysics section, which for the first couple days was done in the kitchen of the Super 8 motel we were all staying at.  That gave everyone the opportunity to interact and argue over the proper way to manipulate tensors.  Come day 4 or so it was all fun and games.

Dinner with Dr. Saulson

The beach was waiting for us once school ended and we usually had a dinner planned with one of the professors.  An Italian guy was fond of gathering up a group to tour the local bars later in the evening, but I was usually too worn out and wouldn’t let myself drink, having hit up the local gym in the afternoon.  Towards the end, we all took the dolphin watch tour, shown below, which was most relaxing.

Dolphin Watch with Dr. Shawhan

As a whole, the program managed to delve into many of the ins and outs of the field, including tidbits from science policy on managing funding for the detectors, the interaction between LIGO and the other facilities such as VIRGO, and the publishing issues that arise in a worldwide collaboration involving thousands of scientists.  Just writing this post reminds me of how excited I was to be there in the company of other students with the same passion as me at the time and with researchers who have no doubt in their minds that a conclusive direct detection of gravitational waves will be made in the next few years.  If professional physics conferences like the one Daniel at CV posted on the other day are anything like this, I see why there are so many of them.  If you are an upper level undergrad or a beginning graduate student with an interest in GWA, consider applying to next year’s summer school.

8.1 Touch-Tone Dialing
Touch-tone telephone dialing is an example of everyday use of Fourier analysis. The
basis for touch-tone dialing is the Dual Tone Multi-Frequency (DTMF) system. The
program touchtone demonstrates how DTMF tones are generated and decoded.
The telephone dialing pad acts as a 4-by-3 matrix (Figure 8.1). Associated with
each row and column is a frequency. These basic frequencies are
fr = [697 770 852 941];
fc = [1209 1336 1477];

tel pad

In order to make a sound in MATLAB, we need to first prompt for the desired number,

%prompts for a keypad # based on its location, i.e. #4 has (k,j)=(2,1)
x(1)=input('enter row k\n');
x(2)=input('enter column j\n');

Those numbers specify the row and column location of the digit, so next we match them to their corresponding frequency according to the dial pad in the picture above. For example, #4 is in the second row and the first column, so fr(k) and fc(j) would respectively give the frequencies 770 and 1209.

%process input
k=x(1);
j=x(2);
fr=[697 770 852 941];
fc=[1209 1336 1477];

Now sound is perceived as a continuous signal. We need it to be discrete in order to digitize it. This is done by defining a large enough sampling rate, which is the number of samples per second to take from the continuous signal and still make it appear smooth. We want to sample for a time long enough to cover the full signal.   A quarter second will do.

Fs=32768; t=0:1/Fs:0.25; %a vector starting at 0, inc by 1/Fs, & ending w/0.25

Say we’re dialing long distance, and the first number we need to enter is “1″.  Here is a plot of what will be sent to the payphone.

Tone

That is generated by

y1=sin(2*pi*fr(k)*t); %vector containing row freq data
y2=sin(2*pi*fc(j)*t); %vector containing column freq data
y=(y1+y2)/2;%average the two; superposition
plot(t,y); %plots y vs. t
axis([0 1/64 -5/4 5/4]) %format: [x_1 x_2 y_1 y_2]
xlabel('t(seconds)')

So now you’ve just fed the phone a mess of a sinusoidal wave.   Let’s hear it:

sound(y,Fs)

Sure, we have a sound; maybe we can hack a pay phone, assuming you can find one.  But so what.  What’s interesting is how the phone will decipher the contents of the tone.  Its electronics will take a Fourier transform of the time signal, a fast and finite Fourier transform (FFT) to be exact, which is an integral that converts the time domain into the frequency domain. Anyone/anything that can perform such an integral deserves to get paid a lot, explaining why pay phones are no longer around and why cell phones are so expensive. The integral involves a product with a complex phase, meaning that as we integrate over time, we circle round and round on the unit circle and pick up a lot of negative components. We therefore want a vector containing the magnitude of the FFT. The plot of this vector of frequency values is known as a periodogram. It is a map of the underlying frequency components of the sinusoidal signal.

The beauty is that you can take a FFT of pretty much anything since every signal can be decomposed into a linear combination of sinusoidal functions by Fourier’s theorem. The value of the FFT on the frequency (horizontal) axis of the periodogram will be zero for all frequencies that are not in the original signal. The height of the nonzero values is proportional to the variance of that frequency component.  It indicates the strength of periodicity at that frequency, as it’s a reflection of the amplitude of the original signal.  In our case, we are only after the frequencies, not the amplitudes, so we can divide by 128 to trim it down a few bits,

y = double(y)/128; %rescales the vector & makes it a dbl
n = length(y); %computes how many elements are y
t = (0:n-1)/Fs; %rescale the time accordingly
p = abs(fft(y)); %computes the magnitude of the fft
f = (0:n-1)*(Fs/n); %sets the freq axis
plot(f,p); %plots the periodogram
axis([500 1700 0 600])

That’s it. The plot should look something like

freq map

and clearly corresponds to “1″ according to the dial pad. You can analyze a full 11-digit signal with Cleve’s touchtone m-file and the help of Chapter 8.

The built-in functions in my opinion make MATLAB a weapon, not a toy.  They are what prompted me to start this blog, as they render accessible a playground on which to extend my imagination, to carry out on a computer screen the phenomena I try to visualize in my head.  They compensate for the time-constraints of a PhD student who would otherwise not have the time to write the necessary code for manipulating data structures that is done with ease in MATLAB.  As processors keep getting faster and faster, speed is becoming less of an issue.  At least in applied mathematics and signal processing applications,  MATLAB is frequently used as the primary computing platform, even in publications.  My guess is that MATLAB will only gain momentum as a means for both teaching and doing physics in the coming years and I hope to stay abreast of the developments.  Hablas MATLAB?

As has become customary in getting started with any program, here goes the appropriate command line in MATLAB:

disp('Hello World!')

…it doesn’t get more simple than that.