Symplectic forms and Darboux coordinates

On a smooth manifold M, a differential two-form is an alternating bilinear pairing on each tangent space, varying smoothly with the point. It is nondegenerate when the only vector whose pairing with every other vector vanishes is zero. In a coordinate chart this means the skew-symmetric coefficient matrix is invertible. An invertible skew matrix has even dimension, so a symplectic manifold has dimension 2n. Equivalently, the top wedge ωⁿ never vanishes. This top form supplies an orientation and the volume form ωⁿ/n!, but still does not define a Riemannian metric.

Closedness is the separate differential condition dω=0. In two dimensions every two-form is automatically closed because there are no nonzero three-forms, so a nowhere-zero area form is symplectic. In higher dimensions, multiplying the standard form by a positive function preserves pointwise nondegeneracy but can destroy closedness. For example, exp(q₁)(dq₁∧dp₁+dq₂∧dp₂) on R⁴ has nonzero derivative exp(q₁)dq₁∧dq₂∧dp₂. Checking only a determinant would therefore miss a genuine failure of the symplectic definition.

Darboux’s theorem says that near every point of a smooth symplectic 2n-manifold there are coordinates (q,p) in which the form is exactly the standard sum. There are no local curvature invariants of the form analogous to the curvature invariants of a Riemannian metric. The theorem does not make a chosen Hamiltonian quadratic, remove singularities of a chosen energy function, or provide one coordinate chart covering the entire manifold. It simplifies the geometric pairing locally while leaving additional structures and global topology to carry information.

A proof uses a local deformation from the given form to its value at the point, then constructs a vector field whose flow corrects that deformation. This is the Moser method; its neighborhood must be small enough for every intermediate form to stay nondegenerate and for the flow to exist over the required parameter interval. We state the general theorem here rather than presenting that full proof. In a surface example, a direct integration of the area density often constructs Darboux coordinates without needing the general deformation machinery.

On a compact symplectic manifold without boundary and of positive dimension, the form cannot be globally exact. If ω=dβ, then ωⁿ=d(β∧ω^{n−1}); Stokes would give zero total integral, contradicting the positive volume determined by ω. This argument explains why local canonical expressions do not imply a global canonical one-form. A sphere with an area form is locally indistinguishable from a planar area form, yet its compact topology prevents one global canonical coordinate chart and one global primitive.

It is also important to distinguish an arbitrary coordinate change from a symplectic transformation of a fixed form. A coordinate chart can absorb a variable coefficient by rewriting the form, whereas a symplectomorphism between two specified symplectic spaces must pull one form back to the other exactly. Area-preserving linear maps in a plane are symplectic, but in dimensions four and higher preserving total volume alone is insufficient. Later lessons will connect this distinction to canonical transformations and the nonsqueezing obstruction, whose content is global rather than visible in a single Darboux chart.