Exercise 2

These are the equations of motion: \[ \dot{p}_1 = -a V^{\prime}(aq_1 -bq_2)\] \[ \dot{p}_2 = b V^{\prime}(aq_1 -bq_2)\]

We multiply the 1st equation by \(b\) and the 2nd equation by \(a\): \[ b\dot{p}_1 = -ab V^{\prime}(aq_1 -bq_2)\] \[ a\dot{p}_2 = ab V^{\prime}(aq_1 -bq_2)\]

Addition of both equations yields: \[ b\dot{p}_1 + a\dot{p}_2 = 0 \Rightarrow \frac{\mathrm{d}}{\mathrm{d}t} (bp_1 + ap_2) = 0 \] \[ \Rightarrow (bp_1 + ap_2) = \text{const.} \]