# Symplectic manifold > [[PORTAL_Dynamical_system|Dynamical system]] spine. ## Microsims — three.js <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Symplectic_manifold.html" width="100%" height="520" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Symplectic manifold — three.js microsim"></iframe> </div> <p class="wt-pending"><strong>Note:</strong> staged, awaiting CDN deploy.</p> A symplectic manifold is a smooth [[Manifold|manifold]] equipped with an extra piece of structure — a closed, non-degenerate 2-form, conventionally written omega — that lets areas, rather than lengths or angles, be measured consistently across the space. Symplectic manifolds are always even-dimensional, and phase space is their canonical example. ## Overview This build renders a symplectic structure directly on a simple phase-space example, showing how the omega form measures oriented area swept out by a system's evolution rather than distance. [[Hamiltonian_mechanics|Hamiltonian mechanics]] is, in modern language, exactly the study of dynamics on a symplectic manifold: Hamilton's equations are the flow generated by the Hamiltonian function through the symplectic structure, and [[Liouville's_theorem_(Hamiltonian)|Liouville's theorem]]'s volume-preservation is a direct consequence of that flow preserving omega itself. The symplectic viewpoint is what lets Hamiltonian mechanics generalise cleanly from a single particle to arbitrarily complicated [[Dynamical_system|dynamical systems]] without changing its basic structure. **On the spine:** [[Dynamical_system]] · [[Manifold]] · [[Hamiltonian_mechanics]] · [[Liouville's_theorem_(Hamiltonian)]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Symplectic_manifold) : [Wikitube](https://en.wikitube.io/wiki/Symplectic_manifold) --- *Repopulated 2026-08-05 · existing three.js asset renamed + wired, Wikipedia-sourced overview · 0 deletions.*