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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{variational bicomplex} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{variational_calculus}{}\paragraph*{{Variational calculus}}\label{variational_calculus} [[!include variational calculus - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{the_bicomplex}{The bicomplex}\dotfill \pageref*{the_bicomplex} \linebreak \noindent\hyperlink{MoreOntheHorizontalComplex}{More on the horizontal differential complex}\dotfill \pageref*{MoreOntheHorizontalComplex} \linebreak \noindent\hyperlink{evolutionary_vector_fields}{Evolutionary vector fields}\dotfill \pageref*{evolutionary_vector_fields} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{horizontal_vertical_and_total_cohomology}{Horizontal, vertical, and total cohomology}\dotfill \pageref*{horizontal_vertical_and_total_cohomology} \linebreak \noindent\hyperlink{the_fundamental_variational_formula}{The fundamental variational formula}\dotfill \pageref*{the_fundamental_variational_formula} \linebreak \noindent\hyperlink{presymplectic_covariant_phase_space}{Presymplectic covariant phase space}\dotfill \pageref*{presymplectic_covariant_phase_space} \linebreak \noindent\hyperlink{symmetries}{Symmetries}\dotfill \pageref*{symmetries} \linebreak \noindent\hyperlink{FormalizationInDifferentialCohesion}{Elementary formalization in differential cohesion}\dotfill \pageref*{FormalizationInDifferentialCohesion} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} For $X$ a ([[spacetime]]) [[manifold]] and $E \to X$ a [[bundle]] (in [[physics]] called the \emph{[[field bundle]]}) with [[jet bundle]] $Jet(E) \to X$, the \textbf{variational bicomplex} is essentially the [[de Rham complex]] $(\Omega^\bullet(Jet( E)),\mathbf{d})$ of $Jet(E)$ with [[differential forms]] $\Omega^n(Jet(E)) = \bigoplus_{h+v=n} \Omega^{h,v}(E)$ bigraded by [[horizontal differential form|horizontal]] degree $h$ (with respect to $X$) and vertical degree $v$ (along the [[fiber]]s of $j_\infty E$)). Accordingly the [[differential]] decomposes as \begin{displaymath} \mathbf{d} = d + \delta \,, \end{displaymath} where $\mathbf{d}$ is the [[de Rham differential]] on $Jet(E)$, $d$ is called the \textbf{horizontal differential} and $\delta$ is called the \textbf{vertical differential}. With $E \to X$ thought of as a [[field bundle]] over [[spacetime]]/[[worldvolume]], then $d$ is a measure for how quantities change over spacetime, while $\delta$ is the [[variational calculus|variational]] differential that measures how quantities change as the field configurations are varied. Accordingly, much of [[classical mechanics]] and [[classical field theory]] on $X$ is formalized in terms of the variational bicomplex. For instance \begin{itemize}% \item a field configuration is a [[section]] of $E$; \item a [[Lagrangian]] is an element $L \in \Omega^{n,0}(E)$; \item a [[local action functional]] is a map \begin{displaymath} S : \Gamma(E) \to \mathbb{R} \end{displaymath} of the form \begin{displaymath} S(\phi) = \int_X L(j^\infty \phi) \,, \end{displaymath} \item the [[Euler-Lagrange equation]] is \begin{displaymath} E(L) := \delta L \mod im d = 0 \end{displaymath} \item the [[covariant phase space]] is the locus \begin{displaymath} \{ \phi \in \Gamma(E) | E(L)(j^\infty \phi) = 0 \} \end{displaymath} \item a [[conserved current]] is an element $\eta\in \Omega^{n-1,0}(E)$ that is horizontally closed on the covariant phase space \begin{displaymath} d \eta = 0 \mod E(L) \end{displaymath} \item a [[symmetry]] is an evolutionary vector field $v$ such that \begin{displaymath} v(L) = 0 \mod im d \end{displaymath} \item [[Noether's theorem]] asserts that every [[symmetry]] induces a [[conserved current]]. \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Let $X$ be a [[smooth manifold]] and $p : E \to X$ some smooth [[bundle]] over $X$. Write $Jet(E) \to X$ for the corresponding [[jet bundle]]. \hypertarget{the_bicomplex}{}\subsubsection*{{The bicomplex}}\label{the_bicomplex} The spaces of [[sections]] $\Gamma(E)$ and $\Gamma(Jet(E))$ canonically inherit a generalized smooth structure that makes them [[diffeological spaces]]: we have a [[pullback]] [[diagram]] of diffeological spaces \begin{displaymath} \itexarray{ \Gamma(E) & \longrightarrow & * \\ \downarrow && \downarrow^{id} \\ [X,E] &\stackrel{p_*}{\longrightarrow}& [X,X] } \,. \end{displaymath} This induces the [[evaluation map]] \begin{displaymath} X \times \Gamma(E) \to E \,. \end{displaymath} and composed with the jet prolongation \begin{displaymath} j^\infty : \Gamma(E) \to \Gamma(Jet(E)) \end{displaymath} it yields a smooth map ([[homomorphism]] of [[diffeological spaces]]) \begin{equation} e_\infty : X \times \Gamma(E) \stackrel{(id,j^\infty)}{\to} X \times \Gamma(Jet(E)) \stackrel{ev}{\to} Jet(E) \,. \label{EProlongation}\end{equation} Write \begin{displaymath} \Omega^{\bullet, \bullet}(X \times \Gamma(E)) \end{displaymath} for the [[cochain complex]] of smooth differential forms on the [[product]] $X \times \Gamma(E)$, bigraded with respect to the differentials on the two factors \begin{displaymath} \mathbf{d} \coloneqq d + \delta \,, \end{displaymath} where the $\mathbf{d}$, $d$ and $\delta$, are the [[de Rham differentials]] of $X\times\Gamma(E)$, $X$ and $\Gamma(E)$, respectively. \begin{defn} \label{}\hypertarget{}{} The \textbf{variational bicomplex} of $E \to X$ is the sub--bi-complex of $\Omega^{\bullet, \bullet}(X \times \Gamma(E))$ that is the [[image]] of the pullback of forms along the map $e_\infty$ \eqref{EProlongation}: \begin{displaymath} e_\infty^* : \Omega^{\bullet}(Jet(E)) \to \Omega^\bullet(X \times \Gamma(E)) \,. \end{displaymath} We write \begin{displaymath} \Omega^{\bullet, \bullet}_{loc} \coloneqq im (e_\infty^*) \end{displaymath} and speak of the bicomplex of \textbf{local forms} on sections on $E$. \end{defn} The bicomplex structure on $\Omega^{\bullet, \bullet}_{loc}$ is attributed in (\hyperlink{Olver86}{Olver 86}) to (\hyperlink{Takens79}{Takens 79}). The above formulation as a sub-bicomplex of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to (\hyperlink{Zuckerman87}{Zuckerman 87, p. 5}). \hypertarget{MoreOntheHorizontalComplex}{}\subsubsection*{{More on the horizontal differential complex}}\label{MoreOntheHorizontalComplex} \begin{remark} \label{CharacterizationOfHorizontalDifferential}\hypertarget{CharacterizationOfHorizontalDifferential}{} In terms of a [[coordinate chart]] $(x^i, u^\alpha,u^\alpha_i,u^\alpha_{i j},\cdots)$ of $E$ covering a coordinate chart $(X^i)$ of $X$, the action of the horizontal differential on functions $f \in C^\infty(Jet(E))$ is given by the formula for the [[total derivative]] operation, but with concrete differentials substituted by the respective jet coordinates: \begin{displaymath} d_h f \;\coloneqq\; \sum_i \left( \frac{\partial f}{\partial x^i} + \frac{\partial f}{\partial u^\alpha}u^\alpha_i + \sum_j \frac{\partial f}{\partial u^\alpha_j} u^\alpha_{j i} + \cdots \right) d x^i \,. \end{displaymath} (\hyperlink{Anderson89}{Anderson 89, p. 10}). \end{remark} More abstractly, the horizontal differential is characterized as follows: \begin{prop} \label{HorizontalDifferentialCompatibleWithPullbackAlongJetProlongations}\hypertarget{HorizontalDifferentialCompatibleWithPullbackAlongJetProlongations}{} The horizontal differential takes horizontal forms to horizontal forms, and for all sections $\phi \in \Gamma(E)$ it respects [[pullback of differential forms]] along the jet prolongation $j_\infty \phi \in \Gamma(Jet(E))$ \begin{displaymath} (j^\infty \phi)^\ast \circ d_h = d \circ (j^\infty \phi)^\ast \end{displaymath} (where on the right we have the ordinary de Rham differential on the base space). \end{prop} More abstractly, the horizontal complex may be understood in terms of [[differential operators]] and the [[jet comonad]] as follows. \begin{remark} \label{HorizontalComplexViaDifferentialOperators}\hypertarget{HorizontalComplexViaDifferentialOperators}{} A horizontal differential $n$-form $\alpha$ on $Jet(E) \to X$ is equivalently a homomorphism of [[bundles]] over $X$ \begin{displaymath} \alpha \colon Jet(E) \longrightarrow \wedge^n T^\ast X \end{displaymath} from the [[jet bundle]] $Jet(E)$ to the [[exterior bundle]] $\wedge^n T^\ast X$. This in turn is, by the discussion there, equivalently a [[differential operator]] $\alpha \colon E \to \wedge^n T^\ast X$. Now of course also the [[de Rham differential]] $d_X$ on $X$ is a differential operator $\wedge^n T^\ast X \to \wedge^n T^\ast X$. In view of this, the horizontal differential of the variational bicomplex is just the [[composition]] operation of differential operators, with horizontal forms regarded as differential operators as above. By the fact that differential operators are the [[co-Kleisli morphisms]] of the [[Jet comonad]], this means that the horizontal differential is \begin{displaymath} d_H \alpha \colon Jet(F) \longrightarrow Jet(Jet(F)) \stackrel{Jet(\alpha)}{\longrightarrow} Jet(\wedge^n T^\ast X) \stackrel{\tilde d_X}{\longrightarrow} \wedge^n T^\ast X \,. \end{displaymath} \end{remark} (e.g. \hyperlink{KrasilshchikVerbovertsky98}{Krasil'shchik-Verbovertsky 98, around def. 3.27}, \hyperlink{KrasilshchikVinogradov99}{Krasil'shchik-Vinogradov 99, ch 4, around def. 1.8}) \hypertarget{evolutionary_vector_fields}{}\subsubsection*{{Evolutionary vector fields}}\label{evolutionary_vector_fields} Vector fields on $J^\infty E$ also split into a direct sum of \textbf{vertical} and \textbf{horizontal} ones, respectively being annihilated by contraction with any horizontal $1$-forms or with any vertical $1$-forms, $\mathfrak{X}(J^\infty E) = \mathfrak{X}_H(J^\infty E) \oplus \mathfrak{X}_V(J^\infty E)$. A special kind of vertical vector field $v \in \mathfrak{X}_V(J^\infty E)$ is called an \textbf{[[evolutionary vector field]]} provided it satisfies $\mathcal{L}_v d = d \mathcal{L}_v$ and $\mathcal{L}_v = \iota_v \delta + \delta \iota_v$, we denote the subspace of evolutionary vector fields as $\mathfrak{X}_{ev}(J^\infty E) \subset \mathfrak{X}_V(J^\infty E)$. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{horizontal_vertical_and_total_cohomology}{}\subsubsection*{{Horizontal, vertical, and total cohomology}}\label{horizontal_vertical_and_total_cohomology} Let $E \to X$ be a smooth [[fiber bundle]] over a base [[smooth manifold]] $X$ of [[dimension]] $n.$ Write $J^\infty E \to X$ for the [[jet bundle]] of $E\to X$. Write \begin{displaymath} \mathcal{F}^s(J^\infty E) \coloneqq I (\Omega^{n,s}(J^\infty E)) \end{displaymath} for the projection of $(n,s)$-forms to the image of the ``interior Euler operator'' (\hyperlink{Anderson89}{Anderson 89, p. 21 (50/318)}). \begin{prop} \label{ELComplexHasSameCohomologyAsDeRhamComplex}\hypertarget{ELComplexHasSameCohomologyAsDeRhamComplex}{} \textbf{(Takens acyclicity theorem)} The [[cochain cohomology]] of the [[Euler-Lagrange complex]] \begin{displaymath} 0 \to \mathbb{R} \to \Omega^{0,0}(J^\infty E) \stackrel{d_H}{\to} \Omega^{1,0}(J^\infty E) \stackrel{d_H}{\to} \cdots \stackrel{d_H}{\to} \Omega^{n,0}(J^\infty E) \stackrel{E}{\to} \mathcal{F}^1(J^\infty E) \stackrel{\delta_V}{\to} \mathcal{F}^2(J^\infty E) \stackrel{\delta_V}{\to} \cdots \end{displaymath} is [[isomorphism|isomorphic]] to the [[de Rham cohomology]] of the total space $E$ of the given fiber bundle. \end{prop} For smooth functions of locally bounded jet order this is due to (\hyperlink{Takens79}{Takens 79}). A proof is also in (\hyperlink{Anderson89}{Anderson 89, theorem 5.9}). For smooth functions of globally bounded order and going up to the Euler-Lagrange operator $E$, this is also shown in (\hyperlink{Deligne99}{Deligne 99, vol 1, p.188}). \hypertarget{the_fundamental_variational_formula}{}\subsubsection*{{The fundamental variational formula}}\label{the_fundamental_variational_formula} \begin{defn} \label{}\hypertarget{}{} A \textbf{source form} is an element $\alpha$ in $\Omega^{n,1}_{loc}$ such that \begin{displaymath} \alpha_\phi(\delta \phi) \end{displaymath} depends only on the 0-jet of $\delta \phi$. \end{defn} \begin{prop} \label{VariationOfLagrangian}\hypertarget{VariationOfLagrangian}{} Let $L \in \Omega^{n,0}_{loc}$. Then there is a unique source form $E(L)$ such that \begin{displaymath} \delta L = E(L) - d \Theta \,. \end{displaymath} Moreover \begin{itemize}% \item $E(L)$ is independent of changes of $L$ by $d$-exact terms: \begin{displaymath} E(L) = E(L + d K) \,. \end{displaymath} \item $\Theta$ is unique up to $d$-exact terms. \end{itemize} \end{prop} This is (\hyperlink{Zuckerman87}{Zuckerman 87, theorem 3}). Here $E$ is the \emph{[[Euler-Lagrange equations|Euler-Lagrange operator]]} . \begin{defn} \label{}\hypertarget{}{} Write \begin{displaymath} \Omega = \delta \Theta \,. \end{displaymath} \end{defn} \begin{remark} \label{HorizontalDOfOmega}\hypertarget{HorizontalDOfOmega}{} By prop. \ref{VariationOfLagrangian} have \begin{displaymath} d \Omega = -\delta E(L) \,. \end{displaymath} \end{remark} \begin{prop} \label{SecondOrderVariationVanishesOnShell}\hypertarget{SecondOrderVariationVanishesOnShell}{} $\delta E$ vanishes when restricted to vertical tangent vectors based in [[covariant phase space]] (but not necessarily tangential to it). \begin{displaymath} \delta E(L) |_{T_{E(L) = 0} \Gamma(E)} = 0 \,. \end{displaymath} \end{prop} This is (Zuckerman 87, lemma 8). \hypertarget{presymplectic_covariant_phase_space}{}\subsubsection*{{Presymplectic covariant phase space}}\label{presymplectic_covariant_phase_space} \begin{cor} \label{}\hypertarget{}{} The form $\Omega$ is a [[conserved current]]. \end{cor} \begin{proof} By remark \ref{HorizontalDOfOmega} and prop. \ref{SecondOrderVariationVanishesOnShell}. \end{proof} \begin{defn} \label{ThePresymplecticForm}\hypertarget{ThePresymplecticForm}{} For $\Sigma \subset X$ a [[compact space|compact]] [[closed manifold|closed]] submanifold of [[dimension]] $n-1$, one says that \begin{displaymath} \omega := \int_\Sigma \Omega \in \Omega^{0,2}_{loc} \end{displaymath} is the [[presymplectic structure]] on [[covariant phase space]] relative to $\Sigma$. \end{defn} \begin{prop} \label{}\hypertarget{}{} The 2-form $\omega$ is indeed closed \begin{displaymath} \delta \omega = 0 \end{displaymath} and in fact exact: \begin{displaymath} \theta := \int_\Sigma \Theta \end{displaymath} is its \emph{presymplectic potential} . \begin{displaymath} \delta \theta = \omega \,. \end{displaymath} \end{prop} \hypertarget{symmetries}{}\subsubsection*{{Symmetries}}\label{symmetries} Let $L \in \Omega^{n,0}_{loc}$. \begin{defn} \label{}\hypertarget{}{} An evolutionary vertical vector field $v \in \mathfrak{X}_{ev}(J^\infty E)$ is a \textbf{[[symmetry]]} if \begin{displaymath} v(L) = 0 \mod im d \,. \end{displaymath} \end{defn} \begin{prop} \label{}\hypertarget{}{} The presymplectic form $\omega$ from def. \ref{ThePresymplecticForm} is annihilated by the Lie derivative of the vector field on $\Gamma(E)$ induced by a symmetry. \end{prop} This appears as (\hyperlink{Zuckerman87}{Zuckerman 87, theorem 13}).0 \hypertarget{FormalizationInDifferentialCohesion}{}\subsection*{{Elementary formalization in differential cohesion}}\label{FormalizationInDifferentialCohesion} We discuss aspects of an [[elementary (infinity,1)-topos|elementary]] formalization in [[differential cohesion]] of the concept of the variational bicomplex . \begin{quote}% under construction \end{quote} Let $\mathbf{H}$ be a context of [[cohesion]] and [[differential cohesion]], with \begin{itemize}% \item [[flat modality]] denoted $\flat$; \item [[infinitesimal shape modality]] denoted $\Im$. \end{itemize} Choose \begin{enumerate}% \item an [[object]] $\Sigma \in \mathbf{H}$, the \emph{base space} (or \emph{[[spacetime]]} or \emph{[[worldvolume]]}); \item an object $E \in \mathbf{H}_{/\Sigma}$, the \emph{[[field bundle]]}, \item an object $\mathbf{A} \in Stab(\mathbf{H}_{/\Sigma}) \stackrel{\Omega}{\to} \mathbf{H}_{/\Sigma}$, the \emph{[[differential cohomology|differential]] [[coefficients]]}. \end{enumerate} Write \begin{itemize}% \item $\mathbf{H}_{/\Sigma} \stackrel{\overset{\sum_\Sigma}{\longrightarrow}}{\stackrel{\overset{\Sigma^\ast}{\longleftarrow}}{\underset{\prod_\Sigma}{\longrightarrow}}} \mathbf{H}$ for the [[base change]] [[adjoint triple]] over $\Sigma$, the [[étale geometric morphism]] of the [[slice (infinity,1)-topos]] $\mathbf{H}_{/\Sigma}$; \item $\Gamma_X \coloneqq \flat \circ \prod_\Sigma \colon \mathbf{H}_{/\Sigma} \to \mathbf{H}$ for the external [[space of sections]] functor; \item $i \colon \Sigma \longrightarrow \Im(\Sigma)$ for the $\Sigma$-component of the [[unit of a monad|unit]] of $\Im$; \item $Jet_\Sigma \coloneqq i^\ast i_\ast$ for the induced [[jet comonad]]; \item $\mathbf{H}_{/\Sigma} \stackrel{\overset{}{\longleftarrow}}{\underset{\iota}{\longrightarrow}} PDE(\mathbf{H})_{\Sigma}$ for the [[Eilenberg-Moore category]] of $Jet_\Sigma$-[[algebras over a monad|coalgebras]] (the objects are [[differential equations]] with [[variables]] in $\Sigma$, the morphisms are [[differential operators]] between these, preserving solution spaces), manifested as a [[topos of coalgebras]] over $\mathbf{H}$; the (non-full) [[direct image]] of this [[geometric morphism]] is the [[co-Kleisli category]] of the [[jet comonad]] and so for $\phi \colon free(E) \to free(F)$ a morphism in $PDE(\mathbf{H})_\Sigma$, we write $\tilde f \colon Jet(E) \to F$ for the corresponding co-Kleisli morphism in $\mathbf{H}_{/\Sigma}$; \end{itemize} We record the following simple fact, which holds generally since the [[jet comonad]] $Jet_\Sigma$ is a [[right adjoint]] (to the [[infinitesimal disk bundle]] functor), hence preserves [[terminal objects]], and $\Sigma \in \mathbf{H}_{/\Sigma}$ is the [[terminal object]]: \begin{prop} \label{JetSigmaPreservesSigma}\hypertarget{JetSigmaPreservesSigma}{} The essentially unique morphism \begin{displaymath} Jet(\Sigma) \stackrel{\simeq}{\longrightarrow} \Sigma \end{displaymath} in $\mathbf{H}_{/\Sigma}$ in an [[equivalence]]. \end{prop} \begin{defn} \label{JetProlongationInDifferentialCohesion}\hypertarget{JetProlongationInDifferentialCohesion}{} The \emph{jet prolongation} map \begin{displaymath} j \colon \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(Jet(E)) \end{displaymath} is the the Jet functor itself, regarded, in view of prop. \ref{JetSigmaPreservesSigma}, as taking [[sections]] to sections via \begin{displaymath} (\Sigma \stackrel{\sigma}{\to} E) \;\;\mapsto \;\; \left( \Sigma \stackrel{\simeq}{\to} Jet(\Sigma) \stackrel{Jet(\sigma)}{\longrightarrow} Jet\left(E\right) \right) \,. \end{displaymath} \end{defn} \begin{defn} \label{HorizontalFormInDifferentialCohesion}\hypertarget{HorizontalFormInDifferentialCohesion}{} For $E \in \mathbf{H}_{\Sigma}$ a [[bundle]] over $\Sigma$, then a \emph{horizontal $\mathbf{A}$-form} on the [[jet bundle]] $Jet(E)$ is a morphism in $PDE(\mathbf{H})_{\Sigma}$ of the form \begin{displaymath} \alpha \colon \iota E \to \iota \mathbf{A} \,. \end{displaymath} For $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A}\to \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism in $\mathbf{PDE}(\mathbf{H})_{\Sigma}$, then the induced \emph{horizontal differential} is the operation of horizontal forms sending $\alpha$ to the composite \begin{displaymath} d \alpha \colon \iota E \stackrel{\alpha}{\longrightarrow} \iota \mathbf{A} \stackrel{d}{\longrightarrow} \iota \mathbf{A}' \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} Since all objects in def. \ref{HorizontalFormInDifferentialCohesion} are in the [[co-Kleisli category]] of the [[jet comonad]], the morphism $\alpha$ there is equivalently a morphism in $\mathbf{H}_{/\Sigma}$ of the form \begin{displaymath} \tilde \alpha \colon Jet(E) \longrightarrow \mathbf{A} \,. \end{displaymath} For the special case that $E = \Sigma$ in def. \ref{HorizontalFormInDifferentialCohesion}, then $Jet_{\Sigma}(\Sigma)\simeq\Sigma$ and so a horizontal $\mathbf{A}$-form on $\Sigma$ we call just a an $\mathbf{A}$-form. \end{remark} \begin{prop} \label{}\hypertarget{}{} The horizontal differential of def. \ref{HorizontalFormInDifferentialCohesion} commutes with pullback of horizontal differential forms $\alpha$ along the jet prolongation, def. \ref{JetProlongationInDifferentialCohesion}, of any field section $\sigma \in \Gamma_X(E)$. In detail: for \begin{itemize}% \item $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A} \longrightarrow \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism, \item $\alpha \colon \iota E \to \iota Et_\Sigma\Sigma^\ast \mathbf{A}$ a horizontal $\mathbf{A}$-form on $Jet(E)$, def. \ref{HorizontalFormInDifferentialCohesion}; \item $\sigma \in \Gamma_\Sigma(E)$ a [[field (physics)|field]] [[section]], \end{itemize} then there is a [[natural equivalence]] \begin{displaymath} j(\sigma)^\ast (d \alpha) \simeq d (j(\sigma)^\ast \alpha) \,. \end{displaymath} \end{prop} \begin{proof} Since all objects are in the [[direct image]] $free\colon \mathbf{H} \to PDE(\mathbf{H})_\Sigma$, this is an equivalence of morphisms in the [[co-Kleisli category]] of the [[jet comonad]], hence is equivalently an equivalence of co-Kleisli composites of morphisms in $\mathbf{H}$. As such, the left hand side of the equality is given in $\mathbf{H}$ by the composite morphism \begin{displaymath} \Sigma \stackrel{\simeq}{\to} Jet(\Sigma) \stackrel{Jet(\sigma)}{\longrightarrow} Jet(E) \stackrel{}{\to} Jet(Jet(E)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\longrightarrow} \mathbf{A}' \,, \end{displaymath} thought of as bracketed to the right. By [[natural transformation|naturality]] of the Jet-counit this is equivalently \begin{displaymath} Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma)) \stackrel{Jet(Jet(\sigma))}{\longrightarrow} Jet(Jet(E)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\longrightarrow} \mathbf{A}' \,, \end{displaymath} By functorality of $Jet(-)$ this is equivalent to \begin{displaymath} Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma)) \stackrel{Jet ( \tilde \alpha \circ Jet(\sigma) )}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\to} \mathbf{A}' \end{displaymath} which is the right hand side of the equivalence to be proven. \end{proof} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[local Lagrangian]], [[Euler-Lagrange form]], [[source form]], [[Lepage form]] \item [[variational calculus]] \item [[secondary differential calculus]] \item [[variational sequence]] \item [[BV-BRST complex]] \item [[BV-BRST variational bicomplex]] \item [[presymplectic current]], [[covariant phase space]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The variational bicomplex was introduced independently in \begin{itemize}% \item [[Alexandre Vinogradov]], \emph{A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints} , Sov. Math. Dokl. 19 (1978) 144--148. \item W. M. Tulczyjew, \emph{The Euler-Lagrange resolution} , in Lecture Notes in Mathematics 836 22--48 (Springer-Verlag, New York 1980). \item T. Tsujishita, \emph{On variation bicomplexes associated to differential equations}, Osaka J. Math. 19 (1982), 311--363. \end{itemize} See also \begin{itemize}% \item [[Floris Takens]], \emph{A global version of the inverse problem of the calculus of variations} J. Diff. Geom. 14 (1979) 543-562 \item [[Pierre Deligne]] et al. \emph{[[Quantum Fields and Strings]]} \end{itemize} An introduction is in \begin{itemize}% \item [[Ian Anderson]], \emph{Introduction to the variational bicomplex}, in \emph{Mathematical aspects of classical field theory}, Contemp. Math. 132 (1992) 51--73, \href{http://books.google.com/books?id=NuiJ0c72_gQC&lpg=PA51&ots=-zAVQLViUn&dq=Zuckerman%20variational%20problems}{gBooks} \end{itemize} A careful discussion that compares the two versions (one over smooth functions globally of finite jet order, one over smooth functions locally of finite jet order) is in \begin{itemize}% \item G. Giachetta, L. Mangiarotti, [[Gennadi Sardanashvily]], \emph{Cohomology of the variational bicomplex on the infinite order jet space}, Journal of Mathematical Physics 42, 4272-4282 (2001) (\href{http://arxiv.org/abs/math/0006074}{arXiv:math/0006074}) \end{itemize} Textbook accounts include \begin{itemize}% \item [[Peter Olver]], section 5.4 of \emph{Applications of Lie groups to differential equations}, Springer Graduate Texts in Mathematics 107 (1986) \item [[Ian Anderson]], \emph{The variational bicomplex}, Utah State University 1989 ([[AndersonVariationalBicomplex.pdf:file]]) \item [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], \emph{Homological Methods in Equations of Mathematical Physics} (\href{http://arxiv.org/abs/math/9808130}{arXiv:math/9808130}) \item [[Joseph Krasil'shchik]], [[Alexandre Vinogradov]] et al. (eds.) \emph{Symmetries and Conservation Laws for Differential Equations of Mathematical Physics}, AMS 1999 \end{itemize} Other surveys include \begin{itemize}% \item Juha Pohjanpelto, \emph{Symmetries, Conservation Laws, and Variational Principles for Differential Equations} (2014) (\href{http://math.usask.ca/~cheviakov/bluman2014/talks/Pohjanpelto.pdf}{pdf slides}) \end{itemize} An early discussion with application to [[covariant phase spaces]] and their [[presymplectic structure]] is in \begin{itemize}% \item [[Gregg Zuckerman|G. J. Zuckerman]], \emph{Action principles and global geometry} , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259\euro{}284. ([[ZuckermanVariation.pdf:file]]) \end{itemize} An invariant version (under group action) is in \begin{itemize}% \item [[Irina Kogan]], [[Peter Olver]], \emph{The invariant variational bicomplex}, \href{http://www4.ncsu.edu/~iakogan/papersPDF/sivKoganOlver.pdf}{pdf} \end{itemize} A more detailed version of this is in \begin{itemize}% \item [[Irina Kogan]], [[Peter Olver]], \emph{Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex}, \href{http://www4.ncsu.edu/~iakogan/papersPDF/ivbKoganOlver-cor.pdf}{pdf} \end{itemize} See also \begin{itemize}% \item [[Victor Kac]], \emph{An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology}, talk at Algebraic Lie Theory, Newton Institute 2009, \href{http://sms.cam.ac.uk/media/538761}{video} \end{itemize} An application to [[multisymplectic geometry]] is discussed in \begin{itemize}% \item Thomas Bridges, Peter Hydon, Jeffrey Lawson, \emph{Multisymplectic structures and the variational bicomplex} (\href{http://personal.maths.surrey.ac.uk/st/T.Bridges/PAPERS/MPCPS-Paper-09024.pdf}{pdf}) \end{itemize} Discussion in the context of [[supergeometry]] is in \begin{itemize}% \item [[Gennadi Sardanashvily]], \emph{Grassmann-graded Lagrangian theory of even and odd variables} (\href{https://arxiv.org/abs/1206.2508}{arXiv:1206.2508}) \end{itemize} [[!redirects horizontal derivative]] [[!redirects horizontal derivatives]] [[!redirects vertical derivative]] [[!redirects vertical derivatives]] [[!redirects variational differential form]] [[!redirects variational differential forms]] [[!redirects total spacetime derivative]] [[!redirects total spacetime derivatives]] [[!redirects horizontal variational complex]] [[!redirects horizontal variational complexes]] \end{document}