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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

$$L = L_1 + L_2 = r_1 \times p_1 + r_2 \times p_2$$

, which is conserved. Mathematically, $L$ could be kept constant while $L_1$ and $L_2$ are changing. Which means we could have motions of the two bodies in two planes at angle to each other, a non-planar motion. Is this allowed in principle? in reality? If not, why? Then, what is reason for the planar motion?

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Yes, take the orbits of Neptune and Pluto, for example, which are inclined to each other.

Furthermore, it is possible for both satellites to influence each other's orbit and in doing so, to run along chaotic trajectories.

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