Definition of Symplectic Manifolds

DefinitionSymplectic Manifold

A symplectic manifold is a pair $(M, \omega)$ where $M$ is a smooth manifold and $\omega$ is a closed, non-degenerate 2-form on $M$ . That is:

Differential form: $\omega \in \Omega^2(M)$ is a smooth 2-form
Closed: $d\omega = 0$
Non-degenerate: For each $p \in M$ , the bilinear form $\omega_p: T_pM \times T_pM \to \mathbb{R}$ is non-degenerate

The form $\omega$ is called the symplectic form or symplectic structure.

The non-degeneracy condition means that at each point $p$ , if $\omega_p(v, w) = 0$ for all $w \in T_pM$ , then $v = 0$ . This makes each tangent space $(T_pM, \omega_p)$ a symplectic vector space, connecting the global manifold structure to the local linear theory.

ExampleStandard Symplectic Structure on $\mathbb{R}^{2n}$

The space $\mathbb{R}^{2n}$ with coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ carries the standard symplectic form:

$\omega_0 = \sum_{i=1}^n dq_i \wedge dp_i$

This is closed ( $d\omega_0 = 0$ since $d$ of a 2-form is a 3-form, and $d^2 = 0$ ) and non-degenerate at every point. The pair $(\mathbb{R}^{2n}, \omega_0)$ is the model for all local symplectic geometry.

Remark

The closedness condition $d\omega = 0$ is crucial: it cannot be deduced from non-degeneracy alone. Closedness ensures that $\omega$ is locally exact (by the Poincaré lemma), allowing us to write $\omega = d\theta$ locally. The 1-form $\theta$ is called a Liouville form or symplectic potential.

Non-degeneracy implies that symplectic manifolds are necessarily even-dimensional: if $\dim M = 2n$ , then $\omega^n = \omega \wedge \cdots \wedge \omega$ ( $n$ times) is a volume form, giving $M$ a canonical orientation and volume.

DefinitionSymplectic Volume Form

On a $2n$ -dimensional symplectic manifold $(M, \omega)$ , the Liouville volume form is:

$\Omega = \frac{\omega^n}{n!}$

This is a nowhere-vanishing top-degree form, making every symplectic manifold orientable with a canonical volume measure.

ExampleCotangent Bundle

Let $Q$ be an $n$ -dimensional manifold. The cotangent bundle $T^*Q$ has a canonical symplectic structure. In local coordinates $(q_i)$ on $Q$ and dual coordinates $(q_i, p_i)$ on $T^*Q$ , the canonical 1-form is:

$\theta = \sum_{i=1}^n p_i dq_i$

The symplectic form is:

$\omega = -d\theta = \sum_{i=1}^n dq_i \wedge dp_i$

This construction is coordinate-independent and provides the natural phase space for classical mechanics on configuration space $Q$ .

The cotangent bundle example is universal: every symplectic manifold is locally symplectomorphic to a cotangent bundle (by Darboux's theorem). This makes $T^*Q$ the fundamental example in symplectic geometry.

Remark

Unlike Riemannian manifolds where the metric varies smoothly, all symplectic forms of the same dimension are locally equivalent. This is the content of Darboux's theorem: around any point $p \in M$ , there exist local coordinates $(q_i, p_i)$ in which $\omega = \sum dq_i \wedge dp_i$ .

DefinitionSymplectomorphism

A diffeomorphism $\phi: (M, \omega) \to (M', \omega')$ between symplectic manifolds is a symplectomorphism if:

$\phi^* \omega' = \omega$

The group of symplectomorphisms of $(M, \omega)$ is denoted $\text{Symp}(M, \omega)$ .

ExamplePhase Space in Classical Mechanics

In Hamiltonian mechanics, the phase space of a system with $n$ degrees of freedom is $\mathbb{R}^{2n}$ (or more generally $T^*Q$ ) with the standard symplectic form. The time evolution preserves this structure: Hamilton's equations define a symplectomorphism for each time value.

For a harmonic oscillator with Hamiltonian $H = \frac{1}{2}(p^2 + q^2)$ , the flow $\phi_t$ given by:

$\phi_t(q, p) = (\cos t \cdot q + \sin t \cdot p, -\sin t \cdot q + \cos t \cdot p)$

is a one-parameter group of symplectomorphisms.