Symplectic geometry studies even-dimensional manifolds carrying a closed, nondegenerate two-form. The form has no notion of length, yet it determines Hamiltonian dynamics, volume, canonical coordinates, and powerful global rigidity.
Symplectic forms and Darboux coordinates
Definition. A symplectic form is \(\omega\in\Omega^2(M)\) such that \[ \mathrm{d}\omega=0,\qquad \omega_p(v,\cdot)=0\Longrightarrow v=0. \] A symplectic manifold is a pair \((M^{2n},\omega)\).
Nondegeneracy forces even dimension and makes \(\omega^n/n!\) a volume form. The model is \[ \omega_0=\sum_{i=1}^n\mathrm{d}q^i\wedge\mathrm{d}p_i \] on \(\mathbb R^{2n}\) or a cotangent bundle.
Theorem (Darboux). Around every point of a symplectic manifold there are coordinates \((q^1,\ldots,q^n,p_1,\ldots,p_n)\) in which \[ \omega=\sum_i\mathrm{d}q^i\wedge\mathrm{d}p_i. \]
Unlike Riemannian metrics, symplectic forms have no local curvature invariants. All of their geometry is global. Moser's path method proves the theorem by solving \(\iota_{X_t}\omega_t=-\alpha_t\) so that the flow of \(X_t\) pulls a varying form \(\omega_t\) back to a fixed one.
Hamiltonian vector fields
Definition. For \(H\in C^\infty(M)\), nondegeneracy defines \(X_H\) by \[ \iota_{X_H}\omega=-\mathrm{d}H. \] In canonical coordinates, \[ \dot q^i=\frac{\partial H}{\partial p_i},\qquad \dot p_i=-\frac{\partial H}{\partial q^i}. \]
Proposition. Hamiltonian flow preserves \(\omega\), the symplectic volume, and \(H\).
Proof. Cartan's formula gives \[ \mathcal L_{X_H}\omega =\mathrm{d}(\iota_{X_H}\omega)+\iota_{X_H}\mathrm{d}\omega =-\mathrm{d}^2H=0. \] Hence the flow preserves \(\omega\) and \(\omega^n\). Moreover, \[ X_HH=\mathrm{d}H(X_H)=-\omega(X_H,X_H)=0. \]
Poisson brackets
Definition. \[ \{F,H\}=\omega(X_F,X_H)=X_HF. \] This bracket is antisymmetric, satisfies Jacobi, and obeys the Leibniz rule. Furthermore, \[ [X_F,X_H]=X_{\{F,H\}}. \]
An observable \(F\) is conserved under \(H\)-flow exactly when \(\{F,H\}=0\). A completely integrable \(2n\)-dimensional system has \(n\) independent commuting integrals; under compactness hypotheses, Arnold-Liouville theory gives action-angle coordinates on invariant tori.
Lagrangian submanifolds
Definition. A submanifold \(L\subset M^{2n}\) is Lagrangian if \(\dim L=n\) and \(\omega|_L=0\).
Example. In \(T^*Q\) with canonical one-form \(\theta=p_i\,\mathrm{d}q^i\) and \(\omega=-\mathrm{d}\theta\), the zero section is Lagrangian. The graph of a one-form \(\alpha\) is Lagrangian exactly when \(\mathrm{d}\alpha=0\), and locally it is the graph of \(\mathrm{d}S\). This is the geometric origin of generating functions and the Hamilton-Jacobi equation.
Symplectic rigidity
Volume preservation is far weaker than symplectic preservation. Gromov's nonsqueezing theorem says that a symplectic ball of radius \(R\) cannot embed into a cylinder \(\{q_1^2+p_1^2<r^2\}\times\mathbb R^{2n-2}\) when \(R>r\), even though the cylinder has infinite volume. Symplectic capacity detects this two-dimensional rigidity inside high-dimensional phase space.