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6. Getting the Math Right

6.1. A Mathematical Proof

Bob: I started out with the recipe provided by Aarseth, Henon and Wielen, and you insisted that we use Binney and Tremaine to check them. We now have five luminaries in stellar dynamics to vouch for our results to be correct. On top of that, we got quite a bit of insight in the underlying physics. Isn't that enough?

Alice: Not to my taste. We have blindly accepted what Binney and Tremaine claim in their appendix, without proving it for ourselves.

Bob: But that is mathematics! Look, there is no physics in the relation between eqs. (62) and (65). It's just a matter of some kind of mathematical transformation, like Fourier transforms or Laplace transforms.

Alice: Even so, we set out to prove things from first principles, and I certainly would feel more comfortable to do exactly that. And I don't feel I can call Abel transforms first principles. Besides, they may well come in handy when we will start constructing more complicated models, and I wouldn't mind getting some more practice in these type of manipulations.

Come on, let's just do it.

Bob: Okay, but not a bit more than is really necessary. Look, they introduce where we only needed the number .

Alice: I'll do it for , why not. You can just watch. So how is the problem posed exactly? In the "if . . then . . " sentence in their appendix . . .

Bob: . . . their "Let . . then . . " sentence . . .

Alice: . . . yes, of course, they are mathematical physicists, not computer scientists. Okay, in the "Let . . then . . " sentence in their appendix, Binney and Tremaine actually assert two results. Let me start trying to prove the first one, which is:



Now how shall we prove this?

6.2. The Problem

Bob: They give a terse hint about substituting the first equation in the second and interchanging the order of the integration. Now why would you want to substitute the first equation in the second? I like brevity, but this is a bit too brief for me.

Alice: Well, given that their book as it is runs already well over 700 pages, they probably thought they couldn't afford more room for explanations.

Bob: Good thing we don't suffer from that constraint. The world wide web is wide enough for our meanderings.

Alice: But let's not meander too far. I want to crack this nut. Let's see. Ah, they must mean that they take the right hand side of eq. (68), and work that out, in order to see whether they really get back in the end.

Bob: What do you mean? They have already called it .

Alice: Well, they have asserted that the right hand side is the answer to the question of what is. We now have to prove it.

Bob: I find that confusing.

Alice: It is a bit confusing. Here, let's be more precise and explicit, without confusing name spaces. Forget about eq. (68) . . .

Bob: . . . with pleasure and glee. I'd just as soon forget about this whole derivation . . .

Alice: Patience, please. Forget about eq. (68), and write instead

Bob: Exactly the same, but now you're calling it .

Alice: Exactly. Exactly exactly, I mean. The point is that now we don't make any claim about what is supposed to mean, it is just a function we happen to define this way, right? Now, this new function also happens to have this factor in the integrand, and nothing stops us from substituting the expression for , given in eq. (67) into eq. (69).

Bob: Yes, and yes. I agree. That is much clearer, since there are only one and one and one in the game, rather than two different with different status. And ah, now I see why you would want to substitute eq. (67) into eq. (69). Is that what they meant?

Alice: I guess it is.

Bob: Okay, now I see why. Boy, they are terse!

Alice: In math, more than half the work is often to find out exactly what to do. Actually doing it is typically the least of the problem.

Bob: But let's do it, and get it over with.

6.3. The Answer

Alice: Yes, let's. Here is the result of substituting eq. (67) into eq. (69):

Bob: What did you do exactly when you went from the second to the third line?

Alice: I followed Binney and Tremaine's advice, to interchange the order of integration.

Bob: But how did you get the new limits for the two integral signs in the third line so quickly?

Alice: If you draw a picture, you can see what is going on here. Instead of use instead. We then have a double integral over the plane. The area over which we take the integral is a triangle bounded on top by the diagonal , on the bottom by the positive axis, and on the right by the line . Now in the second line the inner integral runs over vertical lines, and in the third integral, integration runs over horizontal lines.

[We should probably draw a picture here]

--- picture ---

--- picture ---

--- picture ---

--- picture ---

--- picture ---

Bob: Ah, yes, I see now. It has been a while since I interchanged the order of integrations, I must admit. But your picture makes it clear. Onward!

Alice: Let us focus on the inner integral, and let us call it

It is natural to introduce:

This in turn invites a second change of variables:

That is nice: both the and the dependences have dropped out, and we are left with a definite integral that only has one parameter, .

Bob: And the answer is:

6.4. All the Way

Alice: How did you get that so quickly? Did you use a symbolic integrator? That is cheating?

Bob: No, I used Binney and Tremaine again, their appendix eq. (1B-58). You see, they give you just the minimal amount of information needed to get this all worked out.

Alice: But we haven't worked out the integral yet.

Bob: Are you kidding? Perhaps I should call up a symbolic integrator!

Alice: Look, Bob, we're nearly there, a few feet from the finish line, and you want to give up now? Let's just work it out, and then we can tell all of our friends and family that we have just derived the distribution function corresponding to a softened potential, really from first principles.

Bob: My mom will be thrilled to hear that. I can't get you out of this room until you've proved everything, right? Well, I'll play along under one condition: this time we really will use only . That is all that we needed to prove Binney and Tremaine's main text results, eqs. (52) and (53). Anything more would be masochism.

Alice: Okay, okay, it will be. I'll take a deep breath, and then:

Aha! Now we're getting there. And rather than looking this one up, I do remember how it went. It is all coming back to me now from my freshman years. You start with the following tautology, given that the arcsin function is the inverse of the sin function:

When you differentiate this with respect to , you get

This implies:

This means that the solution of our integral is:

Now that makes me feel good! From first principles, all the way.

6.5. Q.E.D.

Bob: Congratulations with going back to your youth!

Alice: Do I detect a slight sense of sarcasm there?

Bob: Only slight. Next thing you'll do is prove that . Do you have a more first principles all-the-way way of proving that too?

Alice: Well, you start with the empty set, and the notion of a successor mapping, which can be implemented by constructing a set containing the previous number, and then . . .

Bob: . . . I shouldn't have asked!

Alice: Not if you want us to finish today. We're not quite done yet.

Bob: Anyway, I'm glad to see that you're getting at least a wee bit more terse in your derivations, not writing out every change in variables explicitly anymore.

Alice: Yeah, only a wee bit. So. Now we have to substitute our nice result, , for the inner integral in eq. (70). Remembering the original definition of in eq. (69), we can use both of these equations to write, for our :

Quod erat demonstrandum.

Bob: quad what?

Alice: quod, as in quod licet Jovi. Never mind, that is Latin. It means `what was to be demonstrated.' Mathematicians used to write q.e.d at the end of a proof.

Bob: I thought that stood for quantum electrodynamics.

Alice: That too, but we'll keep quantum field theory for volume 137 in our series.

Bob: Remind me, I'm losing track. So you have proved that . Why again did we want to know that?

Alice: You're like my students: lack of motivation leads to loss of memory! We wrote down a definition for the function in (69), and then we substituted the expression within using eq. (62). And then --- but I can see your eyes glazing over. You must be getting tired.

Bob: Too much math, I'm afraid.

Alice: I'll write it down again. We started with

and then we proved that

In words: eq. (79) shows you have to compute , if you start with . Now in order to find the inverse expression, you start instead with , and then you can compute given in (80), and lo and behold, that actually gives you . So the expression between the two equal signs in eq. (80) is the inverse of the expression given in eq. (79): our friend the Abel transform. And the good thing is: we have now proved it completely.

Bob: How nice. And yes, I think you are right: I am getting a bit tired. Can we go home now?

6.6. The End of Let-Then

Alice: Almost. Remember Binney and Tremaine's appendix? Here it is again, there famous "Let . . then . ." claim:



See, we have proved the first equality in the last expression, but we haven't proved the second equality yet. I promise, this will be the last thing we'll do today. I hope it is just a matter of writing things out, since I'm getting a bit tired to, to tell you the truth. One more deep breath, and here we go.

A natural change of variables to simplify the denominator in the integrand for is:

Starting now with the first equality in the last equation above, I'll just see what happens when I do the differentiation with respect to :

Bob: And that is exactly what you wanted to prove. Are you happy now?

Alice: I'm very happy.

Bob: And I even forgot to protest against the fact that you smuggled back into the game again.

Alice: So, we've pulled it off!

Bob: And look how much we have pulled. We have taken a couple sentences from an appendix from Binney and Tremaine, and we have expanded their discussion by a factor of, what? More than ten pages I bet -- an increase of two orders of magnitude! Amazing. No wonder I'm tired. Let's call it a day.

Alice: Fine with me. And tomorrow we'll see your code in action.

Bob: Looking forward to it!
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