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For $X$ a (spacetime) manifold and $E \to X$ a bundle (in physics called the field bundle) with jet bundle $Jet(E) \to X$, the 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 degree $h$ (with respect to $X$) and vertical degree $v$ (along the fibers of $j_\infty E$)). Accordingly the differential decomposes as
where $\mathbf{d}$ is the de Rham differential on $Jet(E)$, $d$ is called the horizontal differential and $\delta$ is called the 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 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
a field configuration is a section of $E$;
a Lagrangian is an element $L \in \Omega^{n,0}(E)$;
a local action functional is a map
of the form
the Euler-Lagrange equation is
the covariant phase space is the locus
a conserved current is an element $\eta\in \Omega^{n-1,0}(E)$ that is horizontally closed on the covariant phase space
a symmetry is an evolutionary vector field $v$ such that
Noether's theorem asserts that every symmetry induces a conserved current.
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.
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
This induces the evaluation map
and composed with the jet prolongation
it yields a smooth map (homomorphism of diffeological spaces)
Write
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
where the $\mathbf{d}$, $d$ and $\delta$, are the de Rham differentials of $X\times\Gamma(E)$, $X$ and $\Gamma(E)$, respectively.
The 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 $e_\infty$ (1):
We write
and speak of the bicomplex of local forms on sections on $E$.
The bicomplex structure on $\Omega^{\bullet, \bullet}_{loc}$ is attributed in (Olver 86) to (Takens 79). The above formulation as a sub-bicomplex of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to (Zuckerman 87, p. 5).
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:
More abstractly, the horizontal differential is characterized as follows:
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 $j_\infty \phi \in \Gamma(Jet(E))$
(where on the right we have the ordinary de Rham differential on the base space).
More abstractly, the horizontal complex may be understood in terms of differential operators and the jet comonad as follows.
A horizontal differential $n$-form $\alpha$ on $Jet(E) \to X$ is equivalently a homomorphism of bundles over $X$
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
(e.g. Krasil’shchik-Verbovertsky 98, around def. 3.27, Krasil’shchik-Vinogradov 99, ch 4, around def. 1.8)
Vector fields on $J^\infty E$ also split into a direct sum of vertical and 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 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 $\mathfrak{X}_{ev}(J^\infty E) \subset \mathfrak{X}_V(J^\infty E)$.
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
for the projection of $(n,s)$-forms to the image of the “interior Euler operator” (Anderson 89, p. 21 (50/318)).
(Takens acyclicity theorem)
The cochain cohomology of the Euler-Lagrange complex
is isomorphic to the de Rham cohomology of the total space $E$ of the given fiber bundle.
For smooth functions of locally bounded jet order this is due to (Takens 79). A proof is also in (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 (Deligne 99, vol 1, p.188).
A source form is an element $\alpha$ in $\Omega^{n,1}_{loc}$ such that
depends only on the 0-jet of $\delta \phi$.
Let $L \in \Omega^{n,0}_{loc}$.
Then there is a unique source form $E(L)$ such that
Moreover
$E(L)$ is independent of changes of $L$ by $d$-exact terms:
$\Theta$ is unique up to $d$-exact terms.
This is (Zuckerman 87, theorem 3).
Here $E$ is the Euler-Lagrange operator .
Write
$\delta E$ vanishes when restricted to vertical tangent vectors based in covariant phase space (but not necessarily tangential to it).
This is ([Zuckerman 87, lemma 8]).
The form $\Omega$ is a conserved current.
For $\Sigma \subset X$ a compact closed submanifold of dimension $n-1$, one says that
is the presymplectic structure on covariant phase space relative to $\Sigma$.
The 2-form $\omega$ is indeed closed
and in fact exact:
is its presymplectic potential .
Let $L \in \Omega^{n,0}_{loc}$.
An evolutionary vertical vector field $v \in \mathfrak{X}_{ev}(J^\infty E)$ is a symmetry if
The presymplectic form $\omega$ from def. 4 is annihilated by the Lie derivative of the vector field on $\Gamma(E)$ induced by a symmetry.
This appears as (Zuckerman 87, theorem 13).0
We discuss aspects of an elementary formalization in differential cohesion of the concept of the variational bicomplex .
under construction
Let $\mathbf{H}$ be a context of cohesion and differential cohesion, with
flat modality denoted $\flat$;
infinitesimal shape modality denoted $\Im$.
Choose
an object $\Sigma \in \mathbf{H}$, the base space (or spacetime or worldvolume);
an object $E \in \mathbf{H}_{/\Sigma}$, the field bundle,
an object $\mathbf{A} \in Stab(\mathbf{H}_{/\Sigma}) \stackrel{\Omega}{\to} \mathbf{H}_{/\Sigma}$, the differential coefficients.
Write
$\mathbf{H}_{/\Sigma} \stackrel{\overset{\sum_\Sigma}{\longrightarrow}}{\stackrel{\overset{\Sigma^\ast}{\longleftarrow}}{\underset{\prod_\Sigma}{\longrightarrow}}} \mathbf{H}$ for the base change adjoint triple over $\Sigma$, the étale geometric morphism of the slice (infinity,1)-topos $\mathbf{H}_{/\Sigma}$;
$\Gamma_X \coloneqq \flat \circ \prod_\Sigma \colon \mathbf{H}_{/\Sigma} \to \mathbf{H}$ for the external space of sections functor;
$i \colon \Sigma \longrightarrow \Im(\Sigma)$ for the $\Sigma$-component of the unit of $\Im$;
$Jet_\Sigma \coloneqq i^\ast i_\ast$ for the induced jet comonad;
$\mathbf{H}_{/\Sigma} \stackrel{\overset{}{\longleftarrow}}{\underset{\iota}{\longrightarrow}} PDE(\mathbf{H})_{\Sigma}$ for the Eilenberg-Moore category of $Jet_\Sigma$-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 $\mathbf{H}_{/\Sigma}$;
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:
The essentially unique morphism
in $\mathbf{H}_{/\Sigma}$ in an equivalence.
The jet prolongation map
is the the Jet functor itself, regarded, in view of prop. 7, as taking sections to sections via
For $E \in \mathbf{H}_{\Sigma}$ a bundle over $\Sigma$, then a horizontal $\mathbf{A}$-form on the jet bundle $Jet(E)$ is a morphism in $PDE(\mathbf{H})_{\Sigma}$ of the form
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 horizontal differential is the operation of horizontal forms sending $\alpha$ to the composite
Since all objects in def. 7 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
For the special case that $E = \Sigma$ in def. 7, then $Jet_{\Sigma}(\Sigma)\simeq\Sigma$ and so a horizontal $\mathbf{A}$-form on $\Sigma$ we call just a an $\mathbf{A}$-form.
The horizontal differential of def. 7 commutes with pullback of horizontal differential forms $\alpha$ along the jet prolongation, def. 6, of any field section $\sigma \in \Gamma_X(E)$.
In detail: for
$d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A} \longrightarrow \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism,
$\alpha \colon \iota E \to \iota Et_\Sigma\Sigma^\ast \mathbf{A}$ a horizontal $\mathbf{A}$-form on $Jet(E)$, def. 7;
then there is a natural equivalence
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
thought of as bracketed to the right. By naturality of the Jet-counit this is equivalently
By functorality of $Jet(-)$ this is equivalent to
which is the right hand side of the equivalence to be proven.
The variational bicomplex was introduced independently in
Alexandre Vinogradov, 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.
W. M. Tulczyjew, The Euler-Lagrange resolution , in Lecture Notes in Mathematics 836 22–48 (Springer-Verlag, New York 1980).
T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311–363.
See also
Floris Takens, A global version of the inverse problem of the calculus of variations J. Diff. Geom. 14 (1979) 543-562
An introduction is in
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
Textbook accounts include
Peter Olver, section 5.4 of Applications of Lie groups to differential equations, Springer Graduate Texts in Mathematics 107 (1986)
Ian Anderson, The variational bicomplex, Utah State University 1989 (pdf)
Joseph Krasil'shchik, Alexander Verbovetsky, Homological Methods in Equations of Mathematical Physics (arXiv:math/9808130)
Joseph Krasil'shchik, Alexandre Vinogradov et al. (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS 1999
Other surveys include
An early discussion with application to covariant phase spaces and their presymplectic structure is in
An invariant version (under group action) is in
A more detailed version of this is in
See also
An application to multisymplectic geometry is discussed in
Discussion in the context of supergeometry is in