Exercise 5

Due to conservation of angular momentum \(mr^2\dot{\Theta} = \text{const.}\) the angular velocity \(\dot{\Theta}\) increases if the radius of the circular motion decreases. This is true, if the Lagrangian does not depend on the angle, e.g.  for a motion with constant velocity on a straight line or the motion with a central force.

But I have to admit that the association between a pendulum of length l and conservation of angular momentum is not clear to me, at least if we assume the existence of a gravitational force. Since the potential energy and therefore the Lagrangian do depend on the angle, angular momentum is not conserved. Of course, without gravitational force angular momentum is conserved, but in this case the particle moves on a circular orbit with radius l. The centripetal force (which is transmitted by the thread of length l) forces the particle to move on a circular orbit. If the length of the thread is decreased, the angular velocitiy will increase. But the motion is not that of a pendulum, which is subject of the exercise.

In Exercise 5, Lecture 7 the Lagrange’s method will be used to obtain the equation of motion of a level pendulum.