Planar motion in central forces.

[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?

 

[A.T.]

why is the solar system flat?

https://www.youtube.com/watch?v=tmNXKqeUtJM

 

[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.

 

[Philip Wood]

precise: are you clearer now?

 

[PJC01]

I can confirm that the motion of two bodies under a central force is indeed restricted to a plane due to the conservation of angular momentum. This is a fundamental law of physics and is based on the principle of conservation of energy and momentum. However, when we consider a system of three bodies, the motion of the other two bodies may not necessarily be restricted to a plane.

The total angular momentum in a three-body system is still conserved, but the individual angular momenta of the two smaller bodies can vary. This means that the motion of the two smaller bodies can occur in two planes that are orthogonal to each other, resulting in a non-planar motion.

The reason why the solar system appears to be flat is due to the initial conditions of its formation. The solar system was formed from a rotating cloud of gas and dust, and as it collapsed under its own gravity, it flattened out into a disk due to conservation of angular momentum. This phenomenon is known as the "conservation of angular momentum in a collapsing cloud" and is the reason why most planetary systems, including our own, are flat.

In summary, the planar motion in central forces is a result of the conservation of angular momentum, which is a fundamental law of physics. However, in a system of three bodies, the motion of the smaller bodies may not be restricted to a plane. The reason for the flatness of the solar system is due to the initial conditions of its formation and the conservation of angular momentum in a collapsing cloud.