Discussion
For a conservative dynamical system, the area enclosed by the phase space :
Answer: B
The area enclosed in phase space remains constant with time.
This is Liouville's theorem. For a Hamiltonian system the flow in phase space is incompressible: the divergence of the phase-space velocity vanishes,
\[ \begin{aligned} &\sum_{i}\left(\frac{\partial\dot{q}_{i}}{\partial q_{i}} + \frac{\partial\dot{p}_{i}}{\partial p_{i}}\right) \\ &= \sum_{i}\left(\frac{\partial^{2}H}{\partial q_{i}\partial p_{i}} - \frac{\partial^{2}H}{\partial p_{i}\partial q_{i}}\right) \\ &= 0 \end{aligned} \]
so a region of phase points may change shape as it moves but keeps its area (volume). Shrinking area belongs to dissipative systems, where trajectories crowd onto attractors, and growing or oscillating area does not occur in Hamiltonian flow.
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