|
|
Randy Poe wrote:
> Ross A. Finlayson wrote:
> > How is it shown that there can be no mathematical closed form solution
> > to an N>2 body problem?
>
> I'm a little fuzzy on the details. Here are a couple of hints:
>
> "Poincaré and Bruns proved that the three-body problem
> is "unsolvable" in the precise sense that it does not
> admit 'enough' analytic integrals to integrate it."
> (http://www.mathaware.org/mam/05/)
>
> "Bruns Theorem: The 10 classical integrals of the three-body
> problem (three for the position of the center of mass, three
> for the velocity of the center of mass, three for the angular
> momentum, and one for the energy) are the only algebraically
> independent integrals of this 18-degree-of-freedom system."
> ( scienceworld.wolfram.com/physics/BrunsTheorem.html">http://scienceworld.wolfram.com/physics/BrunsTheorem.html)
>
> > When I say "closed form solution to a numerical approximation", I mean
> > a closed form solution for the building blocks of the numerical
> > approximation, leading to an exact, one step closed form analytic
> > solution.
>
> If I have n discrete points on a curve, found numerically, it is true
> that I can find a smooth curve that interpolates them, and
> write down a closed-form expression for that curve.
>
> However, there is no sense in which that can be called the
> "solution" to the numerically-derived results. There are
> infinitely many smooth curves which interpolate the same
> points.
>
> > So I just think that there are methods to go from the
> > discrete time serium approximations directly to continuous time
> > approximations.
>
> Precisely. You've just got another approximation. It's silly
> to describe that as "the solution". It's just one more
> approximation.
>
> > Randy, infinite sets are equivalent. Divide by zero.
>
> Non sequitor, and infinite sets are provably not equivalent.
>
> > Anyways, I hope you can enlighten me why there can not be closed form
> > solutions to the three body problem so I can tell the people looking
> > for a theory of everythnig about it.
>
> What does the TOE have to do with the three-body problem?
>
> - Randy
Hi,
Randy, I disagree with Cantor. Do you ascribe zero weight to my
opinion about Cantor? If so, is it s non-zero infinitetesimal weight?
If so, am I right?
Randy, we can talk about the 3-body system with A, B, and C being point
or minutely spherical masses, with A starting at -1, B at 0, and C and
-1, and quickly arrive at analytical solutions. Here's a hint: at
time t, where f(x) is the position of x, f(B) = 0.
Variously, either of A or B can be considered as that the mass of the
other and B is combined and the distance is an easy solution.
There might be elasticity, these are three identical particles, with no
repulsive force the system is solved when f(A)=f(B)=f(C)=0, at a
specific time t, and no further action occurs.
Randy, my friend, please to not be a frickin moron.
Anyways, now I disagree with Poincare, which I thought was pronounced
Pwan-care but which appears to be pronounced Pwon-caray, and Bruns,
which I don't care whether is or not a silent s, and say: non.
Solve the simple 3-body problem. What's the integral over the naturals
of f(x)=1? Why?
Ross
--
Ross A. Finlayson ("Fin le son")
|
|