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Cepterus
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Homework Statement
Given an isolated system of 2 particles in space, we can express the motion of both particles as follows:
$$m_1\ddot{\vec{x_1}}=-\frac\partial{\partial \vec{x_{1}}} V(\vec{x_1},\vec{x_2})\\
m_2\ddot{\vec{x_2}}=-\frac\partial{\partial \vec{x_2}} V(\vec{x_1},\vec{x_2}),$$ where ##V## shall be a potential and ##\frac\partial{\partial \vec{x}}V## shall denote its gradient.
Assuming both particles are at rest at first, prove that they will move on the line which connects both starting points.
Homework Equations
The Attempt at a Solution
In the end, we have to get a result of ##\vec x_i = k_i(\vec x_{2,0} - \vec x_{1,0})##, where ##k_i## is a scalar. Usually I would find ##\vec x## by setting up a differential equation and solving it, but since the equations of motion given here are so general, I don't know how to do this.