Exercise 1

In Lecture 6 the conjugate momentum to \(q_i\) was introduced:

\[ p_i = \frac{\partial L}{\partial \dot{q_i}} \] \[ \Rightarrow \dot{p}_i = \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q_i}} = \frac{\partial L}{\partial q_i}\]

The Lagrangian is: \[ L = \frac{1}{2}( \dot{q}^2_1 + \dot{q}^2_2) - V(q_1-q_2)\]

For the partial derivative we need the chain rule: \[ \frac{\partial L}{\partial q_1} = \frac{\partial L}{\partial V } \frac{\partial V}{\partial (q_1 - q_2)} \frac{\partial (q_1 - q_2)}{\partial q_1} \] \[ = (-1) \cdot V^{\prime}(q_1 - q_2) \cdot 1 \] \[ \Rightarrow \dot{p}_1 = - V^{\prime}(q_1 - q_2) \]

And for \(q_2\): \[ \frac{\partial L}{\partial q_2} = \frac{\partial L}{\partial V } \frac{\partial V}{\partial (q_1 - q_2)} \frac{\partial (q_1 - q_2)}{\partial q_2} \] \[ = (-1) \cdot V^{\prime}(q_1 - q_2) \cdot (-1) \] \[ \Rightarrow \dot{p}_2 = V^{\prime}(q_1 - q_2) \]