Knowlpedia

Theorem

Darboux theorem for symplectic manifolds

Every symplectic form is locally equivalent to the standard constant symplectic form.

Statement

Let (M,ω)(M,\omega) be a 2n2n-dimensional and x∈Mx\in M. The Darboux theorem states that there are

(q1,…,qn,p1,…,pn)(q^1,\ldots,q^n,p_1,\ldots,p_n)

centered at xx in which

ω=∑i=1ndqi∧dpi.\omega=\sum_{i=1}^n dq^i\wedge dp_i.

Equivalently, every point has a neighborhood to an open subset of the standard R2n\mathbb R^{2n}. Hence a symplectic form has no local invariants beyond dimension, although global symplectic manifolds can differ substantially.

Proof idea

Moser's path method interpolates between ω\omega and its constant value at xx, after arranging equality along the chosen point. One solves a time-dependent contraction equation for a and integrates its local flow. The resulting isotopy pulls the interpolating forms back to the initial one; see Cannas da Silva, §1.3.

Consequences

Every symplectic manifold is locally indistinguishable from . In particular, quantities resembling curvature cannot be extracted from a symplectic form alone. Lagrangian and Hamiltonian questions can therefore be studied in standard coordinates locally, while their global gluing and topology retain substantive information.

Conventions and scope

The theorem concerns closed, nondegenerate 22-forms. Nondegeneracy alone does not give this normal form on a neighborhood, because the Moser argument uses closedness. This result is unrelated to about derivatives from real analysis.

References
  1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.3, the Darboux theorem and Moser method.
  2. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Publisher record. Relevant: Chapter 3, local normal forms.