I understand that in a two body problem under central force, corresponding to a potential V(r)(assume one body is massive compared to the other so that its motion is negligible), conservation of angular momentum implies the motion of the body to be in a plane spanned by position r and momentum p vectors.
But if we have three bodies, one of them massive, are the motions of other two bodies still restricted to a plane? Now the total angular momentum is L = L1 + L2 = r1 x p1 + r2 x p2, which is conserved. Mathematically, L could be kept constant while L1 and L2 are changing. Which means we could have motions of the two bodies in two planes orthogonal to each other, a non-planar motion. Is this allowed? If not, why? Then, what is reason for the planar motion?
In specific why is the solar system flat?
But if we have three bodies, one of them massive, are the motions of other two bodies still restricted to a plane? Now the total angular momentum is L = L1 + L2 = r1 x p1 + r2 x p2, which is conserved. Mathematically, L could be kept constant while L1 and L2 are changing. Which means we could have motions of the two bodies in two planes orthogonal to each other, a non-planar motion. Is this allowed? If not, why? Then, what is reason for the planar motion?
In specific why is the solar system flat?
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