Planar motion in central forces.

1. Jun 27, 2014

precise

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?

2. Jun 27, 2014

A.T.

3. Jun 27, 2014

Philip Wood

If the forces on the planets are directed towards a fixed point O, (a good approximation if the Sun is much more massive than any planet), then each planet's angular momentum about O is separately conserved, since if r is the planet's displacement from O, and F is the force on the planet (towards O)…
$$\mathbf r \times \mathbf F = \mathbf r \times (-F\ \mathbf {\widehat{r}}) = 0$$
But…
$$\mathbf r \times \mathbf F = \mathbf r \times \frac{d \mathbf p}{dt} = \frac{d}{dt} (\mathbf r \times \mathbf p) - \frac{d \mathbf r}{dt} \times \mathbf p$$
The last term is zero because
$$\frac{d \mathbf r}{dt} \times \mathbf p = \frac{d \mathbf r}{dt} \times m \frac {d \mathbf r}{dt} = 0$$
So
$$\frac{d}{dt} (\mathbf r \times \mathbf p ) = 0\ \ \ \ \ \ \ \ \text {so} \ \ \ \ \ \ \ \ \mathbf r \times \mathbf p = \mathbf{constant\ vector}$$
Non-mathematically, the argument is simply that for any planet a force towards O can't give rise to a torque about O, so angular momentum about O is conserved.

So orbital planes can be at angles to each other, but in general won't be, because of collisions in the pre-planetary swirl, as A.T.'s excellent clip explains.

Last edited: Jun 28, 2014
4. Jul 1, 2014

Philip Wood

precise: are you clearer now?