Re: n-body problem/variable count

In article <1164570678.977438.187190@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Edward Green" <spamspamspam3@xxxxxxxxxxx> writes:
Dirk Van de moortel wrote:

"Edward Green" <spamspamspam3@xxxxxxxxxxx> wrote in message news:1164564421.822989.26300@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
It is asserted that an n-body problem has 6n variables (3 position and
3 velocity variables for each body). To pursue a common line of
reasoning, we look for constants of the motion, or "first integrals" to
reduce the number of variables. There are 10, and only 10, such
constants, independent of n, and so... well, we really run out of
equations for n= 2, having two degrees of freedom left over: r and w,
for example?

This analysis has the weight of authority, but I'd have to think long
and hard on a first viewing. For, given the 3n position variables for
all time,

For all time?
So you have 3n*infinity positions.
That's slightly more than 3n ;-)

True... the "trajectory function" is in fact infinite dimensional

You know, you may have meant your answer as a joke, but as I think of
this I still haven't hit the understanding minimum. The space of all
feasible trajectories is in fact 6n dimensional... ah, there it is. :-)
It's the intial conditions which are 6n dimensional: we get to place
the n bodies, and we get to give them an initial push. All else is
deterministic. Now, in what sense do we reduce the dimensionality of
the problem through the constants of motion? We still get to set 6n
initial conditions -- the constants more give some additional structure
to the solution space than reduce the number of variables.

Now, another question: the 10 constants of the motion are given as...

center of mass,
linear momentum,
angular momentum,
energy

Oh... OK, I get it. I didn't know what it meant to set the "center of
mass" in general: it's not a conserved quantity. But we can read all
10 as initial conditions <punctuation> we get to set the position of
the center of mass, the linear momentum, the angular momentum... and
the energy? Yes, I guess that's right. Wikipedia seems a little
misleading:

"In other words, integrals provide relations between the variables of
the system, so each scalar integral would normally allow the reduction
of the system's dimension by one unit"

http://en.wikipedia.org/wiki/N-body_problem

I guess that's literally true, but then, there are lots of possible
constraints: these 10 are special. Maybe it's best to think of them as
symmetries?

Yes.

Mati Meron | "When you argue with a fool,
meron@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"
.

Relevant Pages