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**8.8** Two masses \( m_1 \) and \( m_2 \) move in a plane and interact by a potential energy \( U(r) = \frac{1}{2}kr^2 \). Write down their Lagrangian in terms of the CM and relative positions \( \mathbf{R} \) and \( \mathbf{r} \), and find the equations of motion for the coordinates \( X, Y \) and \( x, y \). Describe the motion and find the frequency of the relative motion.

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