Symplectic Geometry
Symplectic 2-forms, Hamiltonian flows, and the geometry of classical phase space.
A symplectic manifold $(M, \omega)$ is an even-dimensional smooth manifold with a closed non-degenerate 2-form $\omega \in \Omega^2(M)$:
$$d\omega = 0, \qquad \omega^n \neq 0 \text{ everywhere} \;\;(\dim M = 2n).$$Canonical example: $T^* Q$ with $\omega = -d\theta$, where $\theta = p_i\, dq^i$ is the Liouville 1-form. In Darboux coordinates,
$$\omega = \sum_{i=1}^n dp_i \wedge dq^i.$$For each smooth function $H: M \to \mathbb R$ (Hamiltonian), the Hamiltonian vector field $X_H$ is defined by $\iota_{X_H}\omega = dH$. Its flow preserves $\omega$ (Liouville's theorem generalizes), and in canonical coordinates reproduces Hamilton's equations $\dot q^i = \partial H/\partial p_i, \dot p_i = -\partial H/\partial q^i$.
Darboux theorem: locally, every symplectic form looks like the standard one — no local invariants (unlike Riemannian geometry, where curvature is local). Global invariants do exist: volume, cohomology class $[\omega] \in H^2$. Gromov's non-squeezing theorem reveals symplectic "width" obstructions invisible to ordinary measure theory.