Static plane in an inertial frame of reference

Question

Suppose we are given a mechanical frame consisting of two points. How can we prove that assuming any initial conditions there is an inertial frame of reference in which these points will be in a static plane?

Answer 1

Here is one interpretation of the question(v1). In the framework of non-relativistic Newtonian mechanics, let us consider an initial frame and two point point masses with initial positions ${\bf r}_1(0)$ and ${\bf r}_2(0)$; and initial velocities ${\bf v}_1(0)$ and ${\bf v}_2(0)$. (By a translation of the initial frame one may assume ${\bf r}_1(0)={\bf 0}$.)

Now perform a Galilean transformation with relative velocity ${\bf v}$ such that the new initial velocities are

$${\bf v}^{\prime}_1(0)~=~{\bf v}_1(0)-{\bf v},$$ $${\bf v}^{\prime}_2(0)~=~{\bf v}_2(0)-{\bf v}.$$

Next find all solutions ${\bf v}$ such that the three initial vectors ${\bf r}_2(0)-{\bf r}_1(0)$, ${\bf v}_1(0)-{\bf v}$, and ${\bf v}_2(0)-{\bf v}$ are linearly dependent. The three linearly dependent vectors therefore span a plane, a line, or a point. (The line and point case will lie inside infinitely many planes.)

Consider finally the full solution of ${\bf v}$'s and planes. Now there should exists at least one such plane so that the total forces ${\bf F}_1(t)$ and ${\bf F}_2(t)$ on each point particle lie in this plane for all times $t$. This plane is going to play the role of OP's static plane, which can be proved with the help of Newton's 2nd law.