6. Getting the Math Right
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
Alice: I'll do it for
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:
Let
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
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
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
Alice: Exactly. Exactly exactly, I mean. The point is that now we
don't make any claim about what
Bob: Yes, and yes. I agree. That is much clearer, since there are only
one
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.
Alice: Yes, let's. Here is the result of substituting
eq. (67) into eq. (69):
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
[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
Bob: And the answer is:
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
Alice: Okay, okay,
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
This implies:
This means that the solution of our integral is:
Now that makes me feel good! From first principles, all the way.
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
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,
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
Alice: You're like my students: lack of motivation leads to loss of memory!
We wrote down a definition for the function
Bob: Too much math, I'm afraid.
Alice: I'll write it down again. We started with
Bob: How nice. And yes, I think you are right: I am getting a bit tired.
Can we go home now?
6.1. A Mathematical Proof
where we only needed the number
.
, why not. You can just watch.
So how is the problem posed exactly? In the "if . . then . . " sentence
in their appendix . . .
6.2. The Problem
back in the end.
.
is. We now have to prove it.
.
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).
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?
6.3. The Answer
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.
and the 
dependences
have dropped out, and we are left with a definite integral that only has
one parameter, 
.
6.4. All the Way
. That is all that we needed to prove Binney
and Tremaine's main text results, eqs. (52) and
(53). Anything more would be masochism.
it will be. I'll take a deep
breath, and then:
, you get
6.5. Q.E.D.
.
Do you have a more first principles all-the-way way of proving that too?
, 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 
:
. Why again did we want to know that?
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.
,
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.
is:
:
back into the game again.