Differential Forms - Applications

Darboux's theorem is the fundamental local structure theorem for symplectic manifolds. It states that all symplectic manifolds locally look the same - there are no local invariants beyond dimension.

TheoremDarboux's Theorem

Let $(M, \omega)$ be a $2n$ -dimensional symplectic manifold. For any point $p \in M$ , there exist local coordinates $(p_1, \ldots, p_n, q_1, \ldots, q_n)$ near $p$ such that

\omega = \sum_{i=1}^n dp_i \wedge dq_i

These are called Darboux coordinates or canonical coordinates.

This remarkable result shows that symplectic geometry has no local invariants - all symplectic manifolds of the same dimension are locally diffeomorphic as symplectic manifolds. Global symplectic geometry, however, is rich with invariants.

Remark

Darboux's theorem contrasts sharply with Riemannian geometry, where the curvature tensor provides local invariants. Symplectic manifolds are "rigid" globally but "flexible" locally.

TheoremMoser's Trick

Let $\omega_t$ be a smooth family of symplectic forms on a compact manifold $M$ with $[\omega_t] = [\omega_0]$ in de Rham cohomology. Then there exists a smooth family of diffeomorphisms $\phi_t: M \to M$ with $\phi_0 = \text{id}$ such that $\phi_t^*\omega_t = \omega_0$ .

Moser's trick is a powerful technique showing that the space of symplectic structures is locally constant in an appropriate sense. It's proven by constructing a time-dependent vector field and integrating its flow.

ExampleSymplectic Linear Algebra

A linear symplectic form on $\mathbb{R}^{2n}$ can be written as $\omega(v,w) = v^T J w$ where

J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}

is the standard symplectic matrix. Linear symplectomorphisms are matrices $A$ with $A^T J A = J$ , forming the symplectic group $\text{Sp}(2n)$ .

TheoremWeinstein's Neighborhood Theorem

Let $L$ be a Lagrangian submanifold of a symplectic manifold $(M, \omega)$ . Then a neighborhood of $L$ in $M$ is symplectomorphic to a neighborhood of the zero section in $T^*L$ with its canonical symplectic structure.

This generalizes Darboux's theorem and provides a canonical model for neighborhoods of Lagrangian submanifolds, fundamental in symplectic topology.

DefinitionPoisson Bracket

On a symplectic manifold $(M, \omega)$ , the Poisson bracket of functions $f, g \in C^\infty(M)$ is

\{f, g\} = \omega(X_f, X_g)

where $X_f$ is the Hamiltonian vector field defined by $i_{X_f}\omega = df$ .

ExampleClassical Mechanics

In phase space $\mathbb{R}^{2n}$ with coordinates $(p_i, q_i)$ and $\omega_0 = \sum dp_i \wedge dq_i$ , the Poisson bracket is

\{f, g\} = \sum_{i=1}^n \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)

Hamilton's equations $\dot{p}_i = -\partial H/\partial q_i$ , $\dot{q}_i = \partial H/\partial p_i$ become $\dot{f} = \{f, H\}$ .

TheoremJacobi Identity

The Poisson bracket satisfies the Jacobi identity:

\{\{f,g\}, h\} + \{\{h,f\}, g\} + \{\{g,h\}, f\} = 0

making $C^\infty(M)$ into a Lie algebra.