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Liouville's Theorem

In Hamiltonian mechanics, the volume occupied by a fluid element remains constant along a trajectory, reflecting the conservation of phase space volume.
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The statement of the theorem

Let x=(q,p)∈R2N\mathbf{x} = (\mathbf{q}, \mathbf{p}) \in \mathbb{R}^{2N} be the phase space coordinates, where q=(q1,…,qN)\mathbf{q} = (q_1, \dots, q_N) and p=(p1,…,pN)\mathbf{p} = (p_1, \dots, p_N). Assume the system evolves according to the Hamiltonian H(q,p)H(\mathbf{q}, \mathbf{p}). The flow x(t)\mathbf{x}(t) is generated by the Hamiltonian vector field v=(q˙,p˙)\mathbf{v} = (\dot{\mathbf{q}}, \dot{\mathbf{p}}), where q˙=∂H/∂p\dot{\mathbf{q}} = \partial H / \partial \mathbf{p} and p˙=−∂H/∂q\dot{\mathbf{p}} = -\partial H / \partial \mathbf{q}. The theorem asserts that the divergence of this vector field vanishes: ∇⋅v=∑i=1N(∂q˙i∂qi+∂p˙i∂pi)=0\nabla \cdot \mathbf{v} = \sum_{i=1}^{N} \left( \frac{\partial \dot{q}_i}{\partial q_i} + \frac{\partial \dot{p}_i}{\partial p_i} \right) = 0 Consequently, the phase space density ρ(x,t)\rho(\mathbf{x}, t) of an ensemble evolving under this flow satisfies the continuity equation: ∂ρ∂t+∑i=1N(∂∂qi(ρq˙i)+∂∂pi(ρp˙i))=0\frac{\partial \rho}{\partial t} + \sum_{i=1}^{N} \left( \frac{\partial}{\partial q_i} (\rho \dot{q}_i) + \frac{\partial}{\partial p_i} (\rho \dot{p}_i) \right) = 0 which implies the conservation of the phase space volume element Ω\Omega: ddt(∫Ωρ(x,t)d2Nx)=0\frac{d}{dt} \left( \int_{\Omega} \rho(\mathbf{x}, t) d^{2N}\mathbf{x} \right) = 0