Exercise 7

In the Lagrangian for the double pendulum, the expression for the potential energy vanishes if there is no gravitational field.

The angular momentum \(p_{\Theta}\) is defined as follows: \[ p_{\Theta} = \frac{\partial L }{\partial \dot{\Theta}} = 3 \dot{\Theta} + \dot{\alpha} + ( 2 \dot{\Theta} + \dot{\alpha}) \cos \alpha \] (see Exercise 6, Lecture 7).

The Lagrangian without gravitational field \[ L = \frac{1}{2} \dot{\Theta}^2 + \frac{\dot{\Theta}^2 + (\dot{\Theta} + \dot{\alpha})^2}{2} + \dot{\Theta} ( \dot{\Theta} + \dot{\alpha} ) \cos \alpha \]

does not depend on \(\Theta\), which is a cyclic coordinate. Hence, the conjugate momentum \(p_{\Theta}\) is conserved: \[ \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L }{\partial \dot{\Theta}} = \frac{\partial L}{\partial \Theta} = 0 \Rightarrow \frac{\partial L }{\partial \dot{\Theta}} = \text{const} \]