Probability Conservation, Liouville Measure, and the Symplectic Origin of Hamiltonian Dynamics
Enmanuel Rodríguez-Brea, Melvin Arias
TL;DR
The paper develops an ensemble-based, coordinate-free foundation for Hamiltonian dynamics by grounding probability transport on the Liouville volume form $Ω=ω^N/N!$ on a symplectic manifold and deriving Liouville’s equation from local conservation. It shows Hamiltonian flows are $Ω$-incompressible, with $\,igl\L_{X_H}ω=0\)$ implying $\bigl\L_{X_H}Ω=0$, and it recovers the canonical equations via $\,\\dot q^i=\\partial H/\\partial p_i$, $\\dot p_i=-\\partial H/\\partial q^i$, while identifying the minus sign with the antisymmetric symplectic structure. The work unifies the hierarchy of conserved quantities through a Taylor expansion of a Liouville-invariant generating function, reveals the Bargmann (central-charge) extension of the Galilean algebra, and develops an extended-phase-space Hamilton–Jacobi framework as an invariance condition. It also clarifies the distinction between total probability conservation and fine-grained information conservation, linking classical symplectic geometry to quantum commutators and offering a pedagogical route that connects probability transport, symplectic structure, and symmetry analysis in a coherent, coordinate-free language.
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 $ρ$ on a $2N$-dimensional symplectic manifold $(\mathcal{M},ω)$, probability transport is expressed intrinsically with respect to the Liouville volume form $Ω=ω^N/N!$ through a continuity equation defined by the $Ω$-divergence. For Hamiltonian evolution, specified by $ι_{X_H}ω=\mathrm{d}H$, Cartan's identity implies $\mathcal{L}_{X_H}ω=0$ and hence $\mathcal{L}_{X_H}Ω=0$, 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 $\{q^i,p_j\}=δ^i_{\ j}$ 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.