Let's solve the problem **step by step** as requested. --- ### **Given:** Hamiltonian: \[ H = \frac{1}{2}p_1^2 + \frac{1}{2}p_2^2 + \frac{1}{2}q_1^2 + \frac{1}{2}q_2^2 \] Poisson bracket: \[ \{f, H\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial H}{\partial q_i} \right) \] Time evolution: \[ \frac{df}{dt} = \{f, H\} \] We are to **find the expression for the generalized momenta \(p_i\) in terms of \(\dot{q}_i\)**. --- ## **Step 1: Hamilton's Equations** Hamilton’s equations relate the generalized coordinates and momenta: \[ \dot{q}_i = \frac{\partial H}{\partial p_i} \] \[ \dot{p}_i = -\frac{\partial H}{\partial q_i} \] --- ## **Step 2: Compute \(\dot{q}_i\) for this Hamiltonian** Let's compute \(\frac{\partial H}{\partial p_i}\): \[ H = \frac{1}{2}p_1^2 + \frac{1}{2}p_2^2 + \frac{1}{2}q_1^2 + \frac{1}{2}q_2^2 \] - For \(i=1\): \[ \frac{\partial H}{\partial p_1} = p_1 \] - For \(i=2\): \[ \frac{\partial H}{\partial p_2} = p_2 \] So, \[ \dot{q}_i = p_i \] --- ## **Step 3: Express \(p_i\) in terms of \(\dot{q}_i\)** From the above: \[ p_i = \dot{q}_i \] --- ## **Step 4: **Final Answer** The expression for the generalized momenta \(p_i\) in terms of \(\dot{q}_i\) is: \[ \boxed{p_i = \dot{q}_i} \] This means **the momentum conjugate to \(q_i\) is simply the time derivative of \(q_i\)** for this system.