 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/2018/s7/3/b/solution

\frac{d F}{d t} = \sum_{i} (F_{q_{i}} \overset{q}{˙}_{i} + F_{p_{i}} \overset{p}{˙}_{i}) = \sum_{i} (F_{q_{i}} H_{p_{i}} - F_{p_{i}} H_{q_{i}}) = [F, H] .

Thus

[F, H] = 0 ⟹ F is conserved

. If an explicitly time-dependent function is allowed, the correct equation is

d F / d t = \partial_{t} F + [F, H]

, so a vanishing bracket alone would not suffice. The source specifies

F (q, p)

, for which that extra term is absent.

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