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I'm having trouble with this question on page 10 of V.I. Arnold's Math Methods book.

A mechanical system consists of three points. At the initial moment their velocities (in some inertial coordinate system) are equal to zero. Show that the points always remain in the plane which contained them at the initial moment.

This should be possible to prove using only the fact that Galilean transformations of space time send solutions of Newton's equation, $\ddot{\bf x}=f(\bf{x}, \dot{\bf{x}})$ to other solutions with different initial conditions.
I'd like to be able to say that $\ddot{\bf x}\cdot (u\times v)=0$ where $u$, $v$ are the relative positions from one particle, centered at the origin, to each of the others (assuming they aren't all three co-linear). Alternatively if I could show that for a rotation $G$ about the normal vector to the plane $(u\times v)$, $G(\ddot{\bf x}\cdot u)=\ddot{\bf x}\cdot u$ then clearly the acceleration is in the plane and the points cannot deviate from it. However I have not been able to make any headway with these attempts, I'd appreciate your suggestions.
Thanks

Corwin Kelly
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  • Here is a nice Arnold style proof for the preceding problem that two points will stay on the line connecting them in the initial moment. This does not immediately solve the current problem but perhaps might be helpful. – Kurt G. Nov 16 '22 at 20:25
  • Thanks @KurtG. I've actually been trying to model a proof of of this one to no avail, it was very helpful in getting a sense of what these proofs might look like but I haven't been able to replicate it for this problem. – Corwin Kelly Nov 17 '22 at 02:49
  • @CorwinKelly In case you are still interested, I have updated the linked answer to make it a full solution, and I believe the strategy will be useful for your question as well. – ummg Jul 26 '23 at 21:41
  • The point is that if you reflect through the plane, both $u$ and $v$ stay the same, so if there was a normal component to the acceleration it would contradict invariance of the acceleration under the action of the reflection. – Vasting Mar 22 '24 at 19:32

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