 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/2017/s7/3/a/solution

(\overset{q}{˙}_{i}, \overset{p}{˙}_{i}) = (H_{p_{i}}, - H_{q_{i}})

\sum_{i} (\partial_{q_{i}} \overset{q}{˙}_{i} + \partial_{p_{i}} \overset{p}{˙}_{i}) = \sum_{i} (H_{q_{i} p_{i}} - H_{p_{i} q_{i}}) = 0.

Thus infinitesimal phase-space volumes are preserved, assuming sufficient differentiability. This is Liouville's theorem; spatial density alone need not be incompressible because the statement concerns the combined coordinate-momentum space.

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