 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012/s7/2/1/solution

For the canonical Poisson bracket, direct substitution of

F = q_{i}

and

F = p_{i}

gives

{q_{i}, G} = \frac{\partial G}{\partial p _{i}}, {p_{i}, G} = - \frac{\partial G}{\partial q _{i}} .

Therefore

\sum_{i} \frac{\partial F}{\partial q _{i}} {q_{i}, G} + \sum_{i} \frac{\partial F}{\partial p _{i}} {p_{i}, G} = \sum_{i} (F_{q_{i}} G_{p_{i}} - F_{p_{i}} G_{q_{i}}) = {F, G} .

This expresses the bracket as a first-order derivative acting on

F

, with coefficients determined by

G

 New to topics? Read the docs here!