# Liouville's theorem (Hamiltonian) > [[PORTAL_Dynamical_system|Dynamical system]] spine. ## Microsims — three.js <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Liouville's_theorem_(Hamiltonian).html" width="100%" height="520" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Liouville's theorem (Hamiltonian) — three.js microsim"></iframe> </div> <p class="wt-pending"><strong>Note:</strong> staged, awaiting CDN deploy.</p> Liouville's theorem, in the Hamiltonian-mechanics sense, states that the phase-space distribution function of a conservative system is constant along the system's trajectories — equivalently, that a region of phase space keeps the same volume as it evolves in time, even as it stretches, twists, and deforms into an increasingly complicated shape. It is named for the French mathematician Joseph Liouville. ## Overview This build renders a small region of phase space as it is carried along by a [[Hamiltonian_mechanics|Hamiltonian]] system's flow, so the volume-preservation can be checked directly even as the region's shape distorts dramatically. The theorem fails the moment friction or any other non-conservative force enters the picture — a damped [[Dynamical_system|dynamical system]]'s phase-space volume shrinks over time as trajectories converge toward an attractor — part of why Liouville's theorem is treated as a defining signature of an idealised conservative system rather than a universal law of motion. It underlies statistical mechanics' assumption that phase-space volume is the natural measure for counting a system's accessible [[Microstate_(statistical_mechanics)|microstates]]. **On the spine:** [[Hamiltonian_mechanics]] · [[Dynamical_system]] · [[Symplectic_manifold]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)) : [Wikitube](https://en.wikitube.io/wiki/Liouville's_theorem_(Hamiltonian)) --- *Repopulated 2026-08-05 · existing three.js asset renamed + wired, Wikipedia-sourced overview · 0 deletions.*