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## 6.1. A Surprise

Carol: Well, let's see what happens. I don't want to look at a thousand lines of output. I will first run the code, redirecting the results into an output file, called euler_try.out:

``` |gravity> ruby euler_try.rb > euler_try.out
```
In that way, we can look at our leisure at the beginning and at the end of the output file, while skipping the 991 lines in between times 0, 0.01, 0.02, 0.03, 0.04 . . . 9.96, 9.97, 9.98, 9.99, 10.

``` |gravity> head -5 euler_try.out
1  0  0  0  0.5  0
1.0  0.005  0.0  -0.01  0.49  0.0
0.9999  0.0099  0.0  -0.0199996250117184  0.480000374988282  0.0
0.999700003749883  0.0147000037498828  0.0  -0.0300001547537506  0.469999845246249  0.0
0.999400002202345  0.0194000022023453  0.0  -0.0400029130125063  0.459997086987494  0.0
|gravity> tail -5 euler_try.out
-19.9935403671885  -16.0135403671885  0.0  -2.24436240147761  -1.74436240147762  0.0
-20.0159839912033  -16.0309839912033  0.0  -2.24435050659793  -1.74435050659793  0.0
-20.0384274962693  -16.0484274962693  0.0  -2.24433863791641  -1.74433863791641  0.0
-20.0608708826484  -16.0658708826485  0.0  -2.24432679534644  -1.74432679534644  0.0
-20.0833141506019  -16.0833141506019  0.0  -2.2443149788018  -1.74431497880181  0.0
```
Dan: A lot of numbers. Now what? We'd better make a picture of the results, to see whether these numbers make sense or not. Let's plot the orbit.

Erica: I agree, we should do that soon. But hey, the numbers do tell us something already, they tell us that there is something seriously wrong!

Carol: How can you tell?

Erica: At the end of the run, the distance between the two particles is more than 25 in our units, as you can see by applying Pythagoras to the last numbers in the first two columns: .

Dan: So what?

Erica: A bit large already for my taste, but what clinches it is the velocity difference between the particles, which is more than .

Dan: So what?

Erica: We started out with a velocity difference of only 0.5, so we have increased the velocity by more than a factor of more than 5, while increasing the distance by a factor of more than 25. When two particles move away from each other, they should slow down, not speed up, because gravity is an attractive force.

Carol: I see, yes, that is strange.

Dan: Even more reason to make a plot!

Carol: How about using gnuplot? That one is present on any system running Linux, and something that can be easily installed on many other systems as well. The style is not particularly pretty, but at least it will give us something to look at.

Dan: How do you invoke gnuplot?

Carol: To use it is quite simple, with only one command needed to plot a graph. In our case, however, I'll start with the command set size ratio -1. A positive value for the size ratio scales the aspect ratio of the vertical and horizontal edge of the box in which a figure appears. But in our case we want to set the scales so that the unit has the same length on both the x and y axes. Gnuplot can be instructed to do so by specifying the ratio to be -1. In fact, you can write the line set size ratio -1 in a file called .gnuplot in your home directory, if you want to avoid repeating yourself each time you use gnuplot. But for starters, I'll give the command specifically.

The next command we need to use is plot which by default will plot the data from the first two columns from the file filename. And of course, you can specify other columns to be used, if you prefer. However, in our case, the first two columns just happen to contain the x and y values of the positions, so there is no need to give any further specifications.

``` |gravity> gnuplot
gnuplot> set size ratio -1
gnuplot> plot "euler_try.out"
gnuplot> quit

```

## 6.2. Too Much, Too Soon

Dan: Hmmm, that is not what I expected to see. What a disappointment!

Erica: Well, research is like that -- the first time you do something, it almost never works.

Dan: It seems as if the system exploded. Why would the two particles fly apart like that?

Erica: That's what we have to find out. And we'd better be systematic.

Dan: How will we ever find out what is the case? Shall we look at the code, line by line, to see whether we made a mistake? It is such a short code, there are not that many ways to do something wrong!

Carol: That's not the right approach. If you are starting from the wrong assumptions, just looking at the code will not help you to realize what was wrong with your thinking, no matter how long you stare at it.

Dan: Research is difficult! If this would be an exercise out of a book, at least the answer would be in the back, or we could ask a teaching assistant . . .

Erica: Yes, research is difficult, but it also is a lot more fun than chewing on home work assignments. You know, when you start playing in your own way, very soon you start doing things that in that exact form nobody else has ever done before. Isn't that a thrill?!

Dan: It would be a thrill if we could make progress. Frankly, I'm lost.

Carol: I must admit, I don't see a clear way ahead either, but at least I remember that one of my teachers told us to `divide and conquer' while troubleshooting. In other words, if something goes wrong in a complex situation, try to simplify everything by dividing whatever procedure you have applied in smaller, more modular steps. That way, you can try to see in which step something goes wrong.

Erica: That makes sense. And I remember hearing a graduate student tell us, while we were struggling with a computer program: `simplify, simplify.' The idea was to first look at the simplest possible parameter choice, because in simpler cases it is often easier to see what goes wrong.

Dan: You mean that we have done too much, too soon, by taking a rather arbitrary choice of initial conditions, and a thousand steps?

Erica: Exactly. The notion of `divide and conquer' tells us that we'd better do one integration step at a time, instead of a thousand. And the idea of `simplify, simplify' suggests that we start with a circular orbit, rather than the more general case of an elliptic orbit.

## 6.3. A Circular Orbit

Carol: So we have to find out what the correct velocity is, for two particles at a distance of 1, in order to move in a circle. It seems to be larger than 0.5, but how much larger?

Dan: Large enough that the particles don't fall toward each other but not so large that they start moving away from each other. Hmm. How can we picture that? Imagine that we would move the two particles in a circular orbit around each other, and measure how much force we have to use to keep them in the circular orbit. We could then require gravity to do the work for us, and insist that the gravitational force would be just equal to the force that we would have to apply by hand.

Erica: Or rather that letting gravity provide the right force, it is easier to compare accelerations, rather than forces. Let us insist on gravity providing the right acceleration.

For the equivalent one-body problem, in our choice of units, the the gravitational acceleration is given in Eq. (42). Since we are only interested in the magnitude, we can write it as:

The acceleration that a particle feels, when being forced to move exactly in a circular orbit is simply given by:

Dan: What do you mean `simply given', how do you know?

Erica: Oh, I just remember, it is one of the standard equations I learned in classical mechanics.

Dan: Well, I don't remember, and while I'm sort-of happy to take your word for it, I would be much happier to see whether we can derive it, so that we know for sure we have the right expression.

Carol: Me too, I'm with Dan here.

Erica: Well, hmmm, I suppose we can go to the library and look it up in a text book on classical mechanics. Any textbook should tell you how to derive that expression. Frankly, I don't remember now how we did it.

Dan: It would be much faster to look it up on Google. But of course, then you have to wonder whether it was done correctly or not.

Carol: Come on, it can't be that hard. And it is much more fun to derive it ourselves rather than look it up. No Dan, I don't even want to look at Google. Here, let's take a piece of paper, and derive both the first and the second derivatives of the scalar distance between the two particles. When we force both derivatives to be zero, we now that will remain constant forever, since equations of motion are second-order differential equations.

Dan: Well, before I ask what you mean, let me first see what you do.

## 6.4. Radial Acceleration

Carol: We start with the definition of as the absolute value or, if you like, the length of the vector :

I will now determine its first time derivative:

On a circular orbit, the distance between the particles is supposed to remain constant, which means , and the only way to guarantee this, according to the equation I just derived, is to insist that the vectors and are perpendicular, so that .

Erica: That makes sense: on a circular orbit the velocity has no component in the direction toward the other particle, so it is indeed perpendicular.

Dan: There is something I don't understand. In the equation above, you start with the expression . But isn't that the velocity? If you insist that , aren't you telling us that the velocity is zero? But in that case the two particles would start falling toward each other, the next moment!

Carol: Which they don't. You are confused with the expression which is the absolute value of the velocity, and it is a very different beast than what I just wrote down. So it is important to realize that, yes, in a one-dimensional situation you can write

but in a two-dimensional or three-dimensional situation this is no longer true in general; in a typical situation we have

Dan: Hmmm. Vector analysis is tricky.

Carol: Until you get used to it.

Dan: Well, that's true for everything.

Carol: Fair enough. Okay, onward to the second derivative of the separation between the two particles:

At the end of the second line I substituted the result of Eq. (52), and at the end of the third line, I used the fact that the position and velocity vector are perpendicular to each other, as we had just derived above. I also used the fact that the acceleration vector points in the opposite direction of the separation vector , which means that .

For a circular orbit, we must insist that the separation between the particles remains constant. This means that the time derivative , and of course the same holds for the second derivative in time, . And there we are, Eq. (55) then gives us:

Dan: Wow, that is exactly the acceleration that Erica remembered, needed to sustain a circular motion.

Erica: Neat! Satisfied, Dan?

Dan: Sure thing!

Carol: Let's see why we did all this. Ah, we wanted to balance the gravitational acceleration provided and the acceleration needed to keep a motion being nicely circular. We already found that , so this means:

or simply:

or equivalently

In our first attempt at orbit integration, we started with an initial condition which implies , but we used an initial velocity of which means that , much too small a value for a circular orbit! It should have been , according to what we just derived.

Dan: Ah, so we should have used for the initial velocity. Great! Good to know.

## 6.5. Virial Theorem

Erica: You know, while Carol was doing her virtuoso derivation act, I suddenly remembered that there is a much quicker way to derive the same result from scratch.

Carol: Show me! I find that hard to believe.

Erica: It just occurred to me that I could use the virial theorem, which tells us that for any bound system, on average the potential energy is equal to minus twice the kinetic energy in the c.o.m. frame. For a circular orbit, both the potential and kinetic energy remain constant, so we don't even have to do any averaging.

In our case, we can use Eqs. (22) and (24) to write the kinetic energy as:

The potential energy is simply:

The virial theorem tells us that , which gives us:

or:

In our units, and therefore we have:

So here you are: for an initial separation of 1, we need an initial velocity of 1.

Carol: I must admit, you got the right answer and your derivation is a bit simpler than the one I just gave. But I have never heard of the virial theorem. What does it mean?

Erica: Weeeeellll, that's quite a long story. I'm not sure whether we should go into that right now. If you really want to know, you can look at a text book, but . . .

Dan: . . . Google gives me a whole bunch of sites. Let's look at a few. Hmmmm. A bit too much math, this one. . . Ah, this one looks easier, with more words and simple examples . . .

Carol: So we know we can look it up when we have to. I agree with Erica, I'd rather move on.

## 6.6. Circular Motion

Dan: Wait a minute, each of you have just given a detailed derivation, and now you're suddenly in a hurry. You know what? I bet that I can give an even simpler derivation, and that without using complicated vector calculus or the vitrial theorem.

Erica: virial theorem.

Dan: Whatever. Here is my suggestion. Why not just write down the circular orbit itself, as if we had already derived it? I don't remember much from my introductory physics class, but I do remember how neat it was that you could write down a simple circular motion in two dimensions in the following way, for the position:

Now this is easy to differentiate. No vector notation, just simple coordinate operations. By differentiation with respect to time, the velocity vectors become:

One more differentiation, and we get the acceleration components:

Comparing Eqs. (65) and (67), we find

We have seen in Eq.(49) that for our initial condition we have , so this means that . Well, Eq. (66) now tells us that . Isn't that simple?

Erica: Yes, it is very simple, I'm surprised!

Dan: I must admit that I'm a bit surprised too, that it came out so easily. And, frankly, I'm surprised that I came out correctly!

Carol: But working in coordinates like that is not very elegant.

Erica: Oh, come on, Carol, give the guy a break! What counts is to get the right answer, and you must admit that his solution is simpler than either of our ways of deriving the same answer. Let's just be glad that all three methods gave the same answer!

Carol: Ah, you physicists, you're so pragmatic! I'd prefer a bit more style.

Dan: Well, each her own style. I'm happy now, and ready to move on!

## 6.7. One Step at a Time

Erica: Which means that we've answered the `simplify, simplify' part of our task of trouble shooting: we now know how to launch the two-body problem on the simplest possible orbit, that of a circle.

The other task was `divide and conquer', and we had already decided to start with just one step.

Dan: That's a simple change in our program: we can just take out the loop.

Carol: Okay, here is the new code. Let me call it euler_try_circular_step.rb.

Dan: You sure like long names! I would have called it euler_trycs.rb.

Carol: Right. And three days later you will be wondering why there is a program floating around in your directory that seems to tell you that it uses Euler's algorithm for trying out cool stuff, or for experimenting with communist socialism or for engaging in some casual sin. No, I'm a big believer in looooong names.

Erica: I used to be like Dan, but I've been bitten too often by the problem you just mentioned, that I could for the life of me not remember what the acronym was supposed to mean that I had introduced. So yes, I'm with you.

Dan: Fine, two against one, I lose again! But I'll be gs, oops, I mean a good_sport.

``` include Math

x = 1
y = 0
z = 0
vx = 0
vy = 1
vz = 0
dt = 0.01

print(x, "  ", y, "  ", z, "  ")
print(vx, "  ", vy, "  ", vz, "\n")

r2 = x*x + y*y + z*z
r3 = r2 * sqrt(r2)
ax = - x / r3
ay = - y / r3
az = - z / r3
x += vx*dt
y += vy*dt
z += vz*dt
vx += ax*dt
vy += ax*dt
vz += az*dt

print(x, "  ", y, "  ", z, "  ")
print(vx, "  ", vy, "  ", vz, "\n")
```

Erica: Let's see: you got the circular velocity correct, a value of unity as it should be. And instead of looping, you print, take one step, and print again. Hard to argue with!

Dan: Let's see whether it gives a reasonable result.
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