A phase domain, its canonical one-form and its standard symplectic two-form

Let $n\ge 1$. A phase domain is a nonempty open set $U\subseteq {\mathbb{R}}^{2n}$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). Its points are written $z=(q,p)$ with $q=(q^1,\dots ,q^n)\in {\mathbb{R}}^n$ and $p=(p_1,\dots ,p_n)\in {\mathbb{R}}^n$; the coordinate functions $q^i,p_i:U\to \mathbb{R}$ are smooth ($C^k$ Euclidean maps and diffeomorphisms), and ${\mathrm{\partial }}_{q^1},\dots ,{\mathrm{\partial }}_{q^n},{\mathrm{\partial }}_{p_1},\dots ,{\mathrm{\partial }}_{p_n}$ is the coordinate frame (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). On a phase domain:

1. the canonical one-form is the smooth one-form ${\lambda }_U:={\sum }_{i=1}^n{p}_{i}d{q}^i\in {\mathrm{\Omega }}^1(U)$;
2. the standard symplectic two-form is the smooth two-form ${\omega }_U:={\sum }_{i=1}^{n}d{q}^i\wedge d{p}_i\in {\mathrm{\Omega }}^2(U)$ (A smooth differential $k$-form);
3. the standard symplectic matrix is the block matrix $J:=\left(\begin{array}{cc}{0}_n& I_n\\ -I_n& 0_n\end{array}\right)\in M_{2n}(\mathbb{R}),$ whose blocks are $n\times n$.

The identity ${\omega }_U=-d{\lambda }_U$ holds, because the coordinate formula for the exterior derivative (The local coordinate formula for the exterior derivative) gives $d{\lambda }_U={\sum }_{i}d{p}_i\wedge d{q}^i=-{\sum }_{i}d{q}^i\wedge d{p}_i$. When $U$ is an induced chart domain of a cotangent bundle, the published Tautological one-form on a cotangent bundle identifies ${\lambda }_U$ and ${\omega }_U$ with the tautological one-form and the canonical cotangent two-form in those coordinates; the present definition is the choice-free local coordinate model that the canonical-calculus items of this page use.