The N-body problem
There are no complete analytical solutions for N>2 body problems.
There are numerical approximations.
We can separate the problem into finding
a) motion of the center of mass;
b) internal motions.
Let’s define center of mass coordinates (like we did for the two body problem):
rki = rk – R , where k = 1, 2, ..., N
R = Smkrk / Smk .
This combination then gives:
Smk rki = 0 , and this leads to
Smk vki = 0 .
As we had with the two body problem, we have:
KE = S½mkvk2 = ½MV2 + S½mkvki 2 ,
P = Smkvk = MV , and
L = Smk (rk ´ vk) = M(R ´ V) + Smk (rki´ vki) .
SFkexternal = MR’’ (but the Fkexternal usually depend on rki )
mk rki’’ = SFkinternal + (SFkexternal - mkR’’) where the last term is usually small.