 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/2015/s7/3/2/solution

\frac{df}{d t} = \sum_{i} (\partial_{q_{i}} f \overset{q}{˙}_{i} + \partial_{p_{i}} f \overset{p}{˙}_{i}) = \sum_{i} (\partial_{q_{i}} f \partial_{p_{i}} H - \partial_{p_{i}} f \partial_{q_{i}} H) = {f, H} .

An explicitly time-dependent function would instead satisfy

df / d t = \partial_{t} f + {f, H}

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