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):

r_kⁱ = r_k – R , where k = 1, 2, ..., N

R = Sm_kr_k / Sm_k .

This combination then gives:

Sm_k r_kⁱ = 0 , and this leads to

Sm_k v_kⁱ = 0 .

As we had with the two body problem, we have:

KE = S½m_kv_k² = ½MV² + S½m_kv_k^{i 2} ,

P = Sm_kv_k = MV , and

L = Sm_k (r_k ´ v_k) = M(R ´ V) + Sm_k (r_kⁱ´ v_kⁱ) .

SF_k^external = MR’’ (but the F_k^external usually depend on r_kⁱ )

m_k r_kⁱ’’ = SF_k^internal + (SF_k^external - m_kR’’) where the last term is usually small.

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