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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 8D / b / Solution

Hamilton's equations and the multivariable chain rule give

\frac{df}{d t} = \sum_{i = 1}^{3} (\frac{\partial f}{\partial q _{i}} \overset{q}{˙}_{i} + \frac{\partial f}{\partial p _{i}} \overset{p}{˙}_{i}) = \sum_{i = 1}^{3} (\frac{\partial f}{\partial q _{i}} \frac{\partial H}{\partial p _{i}} - \frac{\partial f}{\partial p _{i}} \frac{\partial H}{\partial q _{i}}) = {f, H} .

Thus the Hamiltonian conservation law gives

f is conserved ⟺ {f, H} = 0.

Solved by gpt-5.6-sol high.

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