The standard two-form of a phase domain is closed and nondegenerate, with contraction ${\iota }_v\omega ={\sum }_i{a}_{i}d{p}_i-{\sum }_i{b}_{i}d{q}^i$

Let $n\ge 1$, let $U$ be a phase domain with canonical one-form ${\lambda }_U$, standard two-form ${\omega }_U={\sum }_{i}d{q}^i\wedge d{p}_i$ and matrix $J=\left(\begin{array}{cc}{0}_n& I_n\\ -I_n& 0_n\end{array}\right)$ (A phase domain, its canonical one-form and its standard symplectic two-form). For a point $z\in U$ write ${\omega }_z:=({\omega }_U{)}_z$. Then:

1. ${\omega }_U$ is closed: $d{\omega }_U=0$;
2. for all $v,w\in {\mathbb{R}}^{2n}$, written in the coordinate frame as $v=(a,b)$ and $w=(c,d)$ with $a,b,c,d\in {\mathbb{R}}^n$, one has ${\omega }_z(v,w)=v^{\mathsf{T}}Jw=a\cdot d-b\cdot c;$
3. for every $v=(a,b)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ the contraction is the covector ${\iota }_v{\omega }_z={\sum }_{i=1}^n{a}_{i}d{p}_i-{\sum }_{i=1}^n{b}_{i}d{q}^i;$
4. $J^{\mathsf{T}}=-J$, $J^2=-I_{2n}$, and $J$ is invertible with $J^{-1}=J^{\mathsf{T}}=-J$;
5. consequently $v\mapsto {\iota }_v{\omega }_z$ is a linear isomorphism ${\mathbb{R}}^{2n}\to ({\mathbb{R}}^{2n}{)}^{\ast }$, so ${\omega }_U$ is nondegenerate at every point and $(U,{\omega }_U)$ is a symplectic manifold.

• The point of the block form is that all five clauses are pointwise linear-algebra statements with constant coefficients; no completeness, boundedness or topological hypothesis on $U$ is used.
• The published The canonical cotangent two-form is symplectic states the same closedness and nondegeneracy for the canonical form on a cotangent bundle (under countable choice); the present lemma is the choice-free local form on a phase domain, stated so that the flow, canonical-map and generating-function items below can compute with it directly.