Darboux Coordinates and Canonical Form

DefinitionDarboux Coordinates

Let $(M, \omega)$ be a $2n$ -dimensional symplectic manifold. Local coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ on an open set $U \subseteq M$ are called Darboux coordinates (or canonical coordinates) if:

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

In such coordinates, the symplectic form takes the same standard form as on $\mathbb{R}^{2n}$ .

The existence of Darboux coordinates is the content of Darboux's theorem, one of the most important results in symplectic geometry. It shows that symplectic geometry has "no local invariants" — locally, all symplectic manifolds of the same dimension look identical.

ExampleStandard Coordinates on $\mathbb{R}^{2n}$

The standard coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ on $\mathbb{R}^{2n}$ are Darboux coordinates for the standard symplectic form:

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

Any other Darboux coordinates are related to these by a symplectomorphism.

Remark

Darboux coordinates contrast sharply with Riemannian geometry. For a Riemannian manifold $(M, g)$ , one can find normal coordinates at a point where $g_{ij}(p) = \delta_{ij}$ , but the first derivatives $\partial_k g_{ij}$ encode curvature information. In symplectic geometry, Darboux coordinates eliminate all local structure — there is no symplectic curvature.

The non-existence of local invariants makes symplectic geometry fundamentally different from Riemannian geometry. While Riemannian manifolds have local curvature, symplectic manifolds are locally indistinguishable. All the interesting structure in symplectic geometry is global.

DefinitionCanonical Transformation

A diffeomorphism $\phi: (U, \omega) \to (U', \omega')$ between coordinate charts is a canonical transformation if it preserves the symplectic structure:

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

In Darboux coordinates, canonical transformations are precisely the symplectomorphisms.

ExampleCoordinate Changes in Classical Mechanics

In Hamiltonian mechanics, a change of coordinates $(q, p) \mapsto (Q, P)$ is canonical if it preserves Hamilton's equations. This means:

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

Examples include:

Point transformations: $Q_i = Q_i(q), P_i = \sum_j \frac{\partial Q_i}{\partial q_j} p_j$
Generating functions: Transformations defined via $F(q, P)$ with $p_i = \frac{\partial F}{\partial q_i}, Q_i = \frac{\partial F}{\partial P_i}$

DefinitionSymplectic Potential

If $\omega$ is exact on $U$ , we can write $\omega = -d\theta$ for a 1-form $\theta$ . The form $\theta$ is called a symplectic potential or Liouville form.

In Darboux coordinates:

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

The potential is not unique: $\theta' = \theta + df$ for any function $f$ also satisfies $d\theta' = d\theta$ .

Remark

On cotangent bundles $T^*Q$ , the canonical 1-form $\theta = \sum p_i dq_i$ is intrinsic and coordinate-independent. It satisfies $\omega = -d\theta$ globally, not just locally.

ExampleAction Integral

In classical mechanics, the action functional is:

$S[\gamma] = \int_\gamma \theta = \int_\gamma \sum_{i=1}^n p_i dq_i$

Hamilton's principle states that physical trajectories are critical points of the action. The symplectic form $\omega = -d\theta$ encodes the variational structure of mechanics.

DefinitionDarboux Chart

A Darboux chart is a pair $(U, \phi)$ where $U \subseteq M$ is an open set and $\phi: U \to \mathbb{R}^{2n}$ is a diffeomorphism such that:

$\phi^* \omega_0 = \omega|_U$

where $\omega_0$ is the standard symplectic form on $\mathbb{R}^{2n}$ . Darboux's theorem guarantees that every point has a Darboux chart neighborhood.