Integrability of a covariant surface-charge variation

Let $(\mathcal{S},\mathrm{\Omega })$ be exactly a finite-dimensional smooth solution manifold with its smooth closed two-form as specified in Admissible presymplectic reduction and Hamiltonian generators. Let $M$ be the spacetime and $\pi :E\to M$ the chosen field bundle. For a fixed smooth spacetime vector field $\xi$ with local flow ${\mathrm{\Phi }}_t$, suppose a smooth local lift ${\tilde{\mathrm{\Phi }}}_t$ to fibre-preserving local diffeomorphisms of $E$ over ${\mathrm{\Phi }}_t$ is specified, with the local flow composition law. Define its action on a field section by $(A_t\varphi )(x):={\tilde{\mathrm{\Phi }}}_{-t}(\varphi ({\mathrm{\Phi }}_t(x)))\mathrm{.}$ Assume the joint local action $(t,p)\mapsto A_t(p)$ is smooth, preserves the selected solution manifold and boundary conditions, and induces maps $A_t:\mathcal{S}\to \mathcal{S}$. For $p\in \mathcal{S}$, define the induced phase-space vector field by $X_{\xi }(p):={\frac{d}{dt}\mid }_{t=0}{A}_t(p)\in T_p\mathcal{S}\mathrm{.}$ In particular, the association $\xi \mapsto X_{\xi }$ is specified by the lifted action and its preservation hypotheses; it is not inferred merely from the name of a diffeomorphism generator. Define the one-form ${\alpha }_{\xi }$ on $\mathcal{S}$ by $({\alpha }_{\xi }{)}_p(v):={\mathrm{\Omega }}_p(X_{\xi }(p),v),v\in T_p\mathcal{S}\mathrm{.}$ It is integrable on $\mathcal{S}$ when there is a globally defined smooth function $H_{\xi }:\mathcal{S}\to \mathbb{R}$ with $d_{\mathcal{S}}{H}_{\xi }={\alpha }_{\xi }$, using the convention ${\iota }_{X_{\xi }}\mathrm{\Omega }=d_{\mathcal{S}}{H}_{\xi }$. For a smooth map $f:P\to \mathcal{S}$ from a finite-dimensional smooth parameter manifold and a chart $\chi :U\to V\subseteq {\mathbb{R}}^m$ with $V$ open and star-shaped, if the pulled-back one-form $f^{\ast }{\alpha }_{\xi }$ is closed on $U$, then its coordinate representative $({\chi }^{-1}{)}^{\ast }(f^{\ast }{\alpha }_{\xi })$ has a local primitive on $V$. Here $m\ge 1$; if $m=0$, every one-form on the chart is zero and is locally exact. Global integrability on $\mathcal{S}$ is the stronger exactness condition.

Advanced Lectures on General Relativity, §1.4.2

Hamiltonian vector field and Hamiltonian function

Admissible presymplectic reduction and Hamiltonian generators

Closed differential forms are exact on star-shaped coordinate domains

Hamiltonian vector field and Hamiltonian function

The pullback of a differential form

The exterior derivative commutes with pullback

A diffeomorphism direction is null exactly when ${\iota }_{X_{\xi }}\mathrm{\Omega }=0$. If an admissible boundary charge has nonzero variation, that direction is not null and survives a reduction that quotients only by null directions. Neither covariance nor on-shell closure alone guarantees integrability. The potential ambiguity can change boundary terms in $\mathrm{\Omega }$, so integrability must be checked after the boundary conditions and the choice of $\mu$ and $Y$ in the potential ambiguity are specified.