Re: n-body problem/variable count
- From: "Igor" <thoovler@xxxxxxxxxx>
- Date: 27 Nov 2006 10:46:12 -0800
Edward Green wrote:
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?
That's technically true, but easily overcome by realizing that an
elliptical orbit has complete symmetry about the line of apsides, and
that time can be replaced by the orbital angle. Hence the solution can
be written completely in terms of the 10 known integrals.
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, the velocities are completely determined, or else given an
initial position and the velocities for all time, the positions are
determined. So in what sense are there 6n independent variables?
They're considered independent variables in Lagrangian mechanics.
Certainly, if we know x(t) uniquely, we can find dx/dt. But, at the
outset, when we are setting up the problem, we don't know x(t) or even
dx/dt. All we have is a Lagrangian for the system that is a function
of x(t) and dx/dt. This is considered as a function in 6n independent
variables subject to the Euler-Lagrange equations. These equations of
motion, derivable from the Lagrangian, will still need to be solved.
I take it we have recourse to the idea that while knowing the positions
for all time fixes the velocities, knowing them for one time does not,
but I'm not entirely clear on this.
That's correct. One data point doesn't provide much information.
Several of them would be much better, but that still won't tell us much
more about the motion in general. We need to fill in the gaps. And
the only way to actually do this is obtain unique solutions to the
equations.
Also, I wonder why having 8 excess degrees of freedom is qualitatively
worse than having 2: is it obvious we simply won't have families of
solutions in which we get to select another 8 parameters, after
specifying the constants of the motion?
The equations of motion tend to be non-linear and hence extremely
sensitive to small changes in initial data. Thus, extra degrees of
freedom may be covered by observational data, but in a imperfect way.
Periodic corrections are always necessary to keep our caclulations in
sync with nature. Constants of motion are considered superior to
initial conditions, since they do not change within the system. But
there don't seem to any more of them other than those 10.
.
- References:
- n-body problem/variable count
- From: Edward Green
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