A double pendulum consists of two strings of equal length \(\ell\) and two bobs of equal mass \(m\). The upper string is attached to the ceiling, while the lower end is attached to the first bob. One end of the lower string is attached to the first bob, while the other end is attached to the second bob. Using generalized coordinates \(\theta_{1}\) (the angle of the upper string relative to the vertical) and \(\theta_{2}\) (the angle of the lower string relative to the vertical), find

(a) the Lagrangian of the system (Hint: It can be tricky to find the kinetic energy of the lower bob in terms of the angles and their time derivatives. Use Cartesian coordinates initially; then convert these to generalized coordinates)

(b) the canonical momenta

(c) the Hamiltonian in terms of the angles and their first derivatives. Are there any constants of the motion? If so, what are they, and why are they constants? (Note that to go on and find the motion of the system using Hamilton's equations, one must first write \(H\left(\theta_{1}, \theta_{2}, p_{\theta_{1}}\right.\), \(\left.p_{\theta_{1}}\right)\), without \(\dot{\theta}_{1}\) and \(\dot{\theta}_{2}\). This step, and the next step of solving the equations, involves a lot of algebra. This illustrates the fact that in somewhat complicated problems one could long since have written out Lagrange's equations and solved them, by the time one has even written out the Hamiltonian in canonical form.)