Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts

Let $U,V\subseteq {\mathbb{R}}^{2n}$ be phase domains with standard forms and matrix $J$ of A phase domain, its canonical one-form and its standard symplectic two-form. All finite differentiability classes and diffeomorphisms use $C^k$ Euclidean maps and diffeomorphisms.

On any open set $O\subseteq {\mathbb{R}}^m$, a $C^k$ coordinate differential form, $k\ge 0$, is a sum ${\sum }_{i_1<\cdots <i_j}{a}_{i_1\cdots i_j}(z)d{z}^{i_1}\wedge \cdots \wedge d{z}^{i_j}$ with $C^k$ coefficient functions. Its value at a point is the indicated alternating multilinear form on ${\mathbb{R}}^m$. For $k\ge 1$ define its exterior derivative by the coordinate formula $d(ad{z}^{i_1}\wedge \cdots )={\sum }_l({\mathrm{\partial }}_{l}a)d{z}^l\wedge d{z}^{i_1}\wedge \cdots$, extended linearly. For a $C^1$ map $C:U\to V$, the pointwise pullback is $(C^{\ast }\eta {)}_z(v_1,\dots ,v_j):={\eta }_{C(z)}(DC(z)v_1,\dots ,DC(z)v_j)\mathrm{.}$ It is a continuous form when $\eta$ is continuous; no assertion that this pullback is smooth is part of the definition. Wedges and contractions are the pointwise alternating multilinear operations. Thus the finite-regularity identities below can be read without applying a definition restricted to smooth forms or maps.

A $C^r$ canonical map, $1\le r\le \mathrm{\infty }$, is a $C^r$ map $C:U\to V$ satisfying $C^{\ast }{\omega }_V={\omega }_U$ pointwise. A $C^r$ canonical transformation is such a map that is a $C^r$ diffeomorphism onto $V$; a local $C^r$ canonical transformation is a $C^r$ local diffeomorphism satisfying the same equality. The $C^{\mathrm{\infty }}$ case agrees with Canonical transformation and Symplectomorphisms, local symplectomorphisms, and symplectic embeddings; the finite-$r$ terminology on this page is the definition just given, and does not assert that a $C^1$ map is smooth.

For $h\in C^2(U)$ the coordinate Hamiltonian field is the $C^1$ vector-valued map $X_h:=J\mathrm{\nabla }h$, viewed as a field by the canonical identification $T_{z}U={\mathbb{R}}^{2n}$. Its characterization by ${\iota }_{X_h}{\omega }_U=dh$ is proved in The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$ ↗. For $f,g\in C^1(U)$ define the coordinate Poisson bracket by $\{f,g\}:=\mathrm{\nabla }{f}^{\mathsf{T}}J\mathrm{\nabla }g$; for $C^2$ inputs it is $C^1$ and equals ${\omega }_U(X_f,X_g)$ by that proposition. Smooth inputs recover the smooth symplectic conventions. No closure of the $C^2$ class under this bracket is assumed.