Two blocks of equal mass \(m\), connected by a Hooke's-law spring of unstretched length \(\ell\), are free to move in one dimension. Find the equations of motion of the system, using the relative and center of mass coordinates introduced in the preceding problem.

Data from Problem 4.20

Center of mass and relative coordinates. Show that for two particles moving in one dimension, with coordinates \(x_{1}\) and \(x_{2}\), with a potential that depends only upon their separation \(x_{2}-x_{1}\), then the Lagrangian

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can be rewritten in the form 1 L = mx + mx - U(x2 x1) - L = 1 = x + x - U(x), where Mm + m2 is the total mass and = m1m2/M is the "reduced mass" of the system, and X (mx+m2x2)/M is the center of mass coordinate and x' = x2-x is the relative coordinate. =