8.9 ** Consider two particles of equal masses, m₁ = m2, attached to each other by a light straight spring (force constant k, natural length L) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r₁ and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r, ) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X, Y. (c) Write down the Lagrange equations for r and p. Solve these for the two special cases that r remains constant and that remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is w=√2k/m1.

Fig: 1