 Solution

ID: past-exam-of-the-classical-mechanics-course-of-the-university-of-oxford/2023/3/a/ii/solution

{f, g} = i = 1 \sum n (\frac{\partial f}{\partial q _{i}} \frac{\partial g}{\partial p _{i}} - \frac{\partial f}{\partial p _{i}} \frac{\partial g}{\partial q _{i}}) .

It treats

t

\frac{df}{d t} = \sum_{i} (f_{q_{i}} \overset{q}{˙}_{i} + f_{p_{i}} \overset{p}{˙}_{i}) + f_{t} = \sum_{i} (f_{q_{i}} H_{p_{i}} - f_{p_{i}} H_{q_{i}}) + f_{t} .

\frac{df}{d t} = {f, H} + \frac{\partial f}{\partial t} .

The total derivative follows the trajectory; the explicit partial derivative holds

(q, p)

fixed.

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