Probability Conservation, Liouville Measure, and the Symplectic Origin of Hamiltonian Dynamics

Abstract

Liouville's theorem -- the preservation of phase-space volume -- is often presented as a corollary of Hamilton's canonical equations. Here we adopt an ensemble-first viewpoint in which the starting point is local probability conservation on phase space. For a probability density $\rho$ on a $2N$-dimensional symplectic manifold $(\mathcal{M},\omega)$, probability transport is expressed intrinsically with respect to the Liouville volume form $\Omega=\omega^N/N!$ through a continuity equation defined by the $\Omega$-divergence. For Hamiltonian evolution, specified by $\iota_{X_H}\omega=\mathrm{d}H$, Cartan's identity implies $\mathcal{L}_{X_H}\omega=0$ and hence $\mathcal{L}_{X_H}\Omega=0$, so the Hamiltonian flow is incompressible in the Liouville sense and the continuity law reduces to Liouville's equation. In canonical coordinates this reproduces Hamilton's equations. In particular, the canonical Poisson-bracket relations $\{q^i,p_j\}=\delta^i_{\ j}$ provide the kinematic input that fixes the evolution of observables and underlies the canonical form of the continuity equation. The same organization clarifies the distinction between conservation of total probability and preservation of fine-grained information measures (Gibbs--Shannon entropy), which holds specifically for Liouville-measure-preserving dynamics.