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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?

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Welcome to Physics.SE. I have to say, I've read this question a couple of times and can't figure out what you're asking here. Can you be more explicit? Or possibly link to an image (so that someone can edit it into your question--you'll be able to put them in yourself once you have a little rep)? – dmckee Feb 3 '12 at 2:11
Is this a homework question? – Harry Johnston Feb 3 '12 at 3:35
Can you be more precise when you say "...assuming any initial conditions..." ? Are rotations allowed? If so then I dont understand how you can find an "inertial frame of reference" in which the points are static. – Vijay Murthy May 3 '12 at 14:20

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.

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