Two point masses \(m\) and \(M\) are connected by a massless rod of length \(\ell\) and placed on a horizontal table. The mass \(m\) is also connected to a fixed point \(P\) on the table by a spring with spring constant \(k\) and zero natural length; it can slide without friction in any direction on the table. The mass \(M\) is constrained to slide along a straight line at a perpendicular distance \(b<\ell\) from \(P\). Set up a Lagrangian for the system and obtain the Euler-Lagrange equations. In the limit of small oscillations, solve for the characteristic angular frequencies and normal modes. Explain whether or not energy is conserved.