Establish Lagrange's equations of motion for a general dynamical system, including the form they reduce to for impulsive actions. Give proofs of the following general theorems:
[(i)] The reciprocal relation \(\dot{q}_s/P_r = \dot{q}'_r/P'_s\).
[(ii)] Bertrand's Theorem that the energy generated by given impulses is diminished by the introduction of any constraint.
[(iii)] Kelvin's Theorem that the energy acquired when the system is started with prescribed velocities is increased by the introduction of any constraint.
Illustrate these theorems by discussing the motion of a waggon of mass M carrying a pendulum of mass m and length l which can swing in the direction of motion of the waggon.