1. A particle of mass m slides along the inside of a smooth spherical bowl of radius r. The particle remains in a vertical plane (the xz-plane). (a) Assuming that the bowl does not move: (i) Write down the Lagrangian, taking the angle 0 with respect to the vertical direction as the generalized coordinate. (ii) Hence, derive the equation of motion for the particle. (b) Assuming that the bowl rests on a smooth horizontal table and has a mass M, and that the bowl can slide freely along the x-direction: (i) Write down the Lagrangian in terms of the angle 0 and the x-coordinate of the bowl, x. (ii) Hence, derive the equations of motion for the system. (iii) Identify any cyclic coordinates in the system and interpret the corresponding conserved momenta. (iv) Using the expressions for the conjugate momenta, determine expressions for the generalized velocities in terms of the generalized coordinates and their conjugate momenta.

1. A particle of mass m slides along the inside of a smooth spherical bowl of radius r. The particle remains in a vertical plane (the xz-plane). (a) Assuming that the bowl does not move: (i) Write down the Lagrangian, taking the angle 0 with respect to the vertical direction as the generalized coordinate. (ii) Hence, derive the equation of motion for the particle. (b) Assuming that the bowl rests on a smooth horizontal table and has a mass M, and that the bowl can slide freely along the x-direction: (i) Write down the Lagrangian in terms of the angle 0 and the x-coordinate of the bowl, x. (ii) Hence, derive the equations of motion for the system. (iii) Identify any cyclic coordinates in the system and interpret the corresponding conserved momenta. (iv) Using the expressions for the conjugate momenta, determine expressions for the generalized velocities in terms of the generalized coordinates and their conjugate momenta.