Lagmental Vicfred

A Symplectic Form Is Closed and Nondegenerate by Vicfred

Last updated: Thu 12 October 2017

tags: mathematics, symplectic geometry, Hamiltonian systems, Morse theory

The top wedge power of a symplectic form is a volume form. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Start locally

A symplectic manifold \((M^{2n},\omega)\) has a closed nondegenerate two-form. A Hamiltonian \(H:M\to\mathbf R\) determines a vector field through contraction with \(\omega\).

$$ d\omega=0,\qquad v\mapsto\iota_v\omega\ \text{is an isomorphism} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \omega^n=\underbrace{\omega\wedge\cdots\wedge\omega}_{n\text{ factors}}\ne0 $$

Compute before generalising

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \omega_0=\sum_{i=1}^{n}dq_i\wedge dp_i,\qquad\frac{\omega_0^n}{n!}=dq_1\wedge dp_1\wedge\cdots\wedge dq_n\wedge dp_n $$

The global view

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad d\omega=0,\qquad v\mapsto\iota_v\omega\ \text{is an isomorphism},\\[5pt] \mathsf{C}\;&:\quad \omega^n=\underbrace{\omega\wedge\cdots\wedge\omega}_{n\text{ factors}}\ne0. \end{aligned} $$

Edge conditions

Symplectic geometry has no preferred notion of distance. Nondegeneracy of a two-form is not positive definiteness of a metric.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \omega^n=\underbrace{\omega\wedge\cdots\wedge\omega}_{n\text{ factors}}\ne0 \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Thu 08 May 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.