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For $X$ a (spacetime) manifold and $E \to X$ a bundle (in physics called the field bundle) with jet bundle $j_\infty E \to X$, the variational bicomplex is essentially the de Rham complex $(\Omega^\bullet(j_\infty E),\mathbf{d})$ of $j_\infty E$ with differential forms $\Omega^n(j_\infty E) = \bigoplus_{h+v=n} \Omega^{h,v}(E)$ 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 $j_\infty E$, $d$ is called the horizontal differential and $\delta$ is called the vertical differential.
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 a vertical 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 $j_\infty E \to X$ for the corresponding jet bundle.
The spaces of sections $\Gamma(E)$ and $\Gamma(j_\infty E)$ 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) to (Takens). The above formulation as the image of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to (Zuckerman, p. 5).
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, 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, 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 $L \in \Omega^{n,0}_{loc}$.
A vertical vector fields $v \in T_v E \subset T_v j_\infty E$ is a symmetry if
The presymplectic form $\omega$ from def. 4 is degenerate on symmetries.
This appears as (Zuckerman, theorem 13).
The variational bicomplex was introduced independently in
W.M. Tulczyjew, The Euler-Lagrange resolution , in Lecture Notes in Mathematics 836 22–48 (Springer-Verlag, New York 1980).
A.M. 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.
A.M. Vinogradov, The $C$-spectral sequence, Lagrangian formalism and conservation laws I, II, J. Math. Anal. Appl. 100 (1984) 1–129.
Also
An textbook account is in section 5.4 of
An invariant version (under group action) is in
An early discussion with application to covariant phase spaces and their presymplectic structure is in
An introduction is in
Ian Anderson, Introduction to the variational bicomplex in Mathematical aspects of classical field theory, Contemp. Math. 132 (1992) 51–73, gBooks
Ian Anderson, The variational bicomplex, Utah State University (1989) (pdf) (textbook account)
Victor Kac, An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology, talk at Algebraic Lie Theory, Newton Institute 2009, video
An application to multisymplectic geometry is discussed in