Given a smooth bundle $E \to \Sigma$ over a smooth manifold $\Sigma$, then its Euler-Lagrange complex is a resolution of the constant sheaf of locally constant functions on the jet bundle $J^\infty E$ by a chain complex of sheaves of certain differential forms. The Euler-Lagrange complex starts out as the complex of horizontal differential forms up to degree $n \coloneqq dim(\Sigma)$ the dimension of $\Sigma$, the following differential is
the Euler-Lagrange operator $\delta_{El}$
followed by the Helmholtz operator $\delta_{Helm}$
Hence the elements in the Euler-Lagrange complex have the following interpretation
in degree $dim(\Sigma)$: local Lagrangians;
in degree $dim(\Sigma)+1$: Euler-Lagrange equations of motion;
in degree $dim(\Sigma)-1$: trivial local Lagrangians;
in degree $dim(\Sigma)+2$: obstructions for equations of motion to be variational, i.e. to be the Euler-Lagrange equations of a local Lagrangian.
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.
The Euler-Lagrange complex was recognized in
Alexandre Vinogradov, A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978), 144–148.
W. M. Tulczyjew, The Euler-Lagrange resolution, in Lecture Notes in Mathematics No. 836, Springer-Verlag, New York, 1980, pp. 22–48.
Review includes
Alexandre Vinogradov, I. S. Krasilshchik (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. (pdf)
Ian Anderson, The variational bicomplex, Utah State University 1989 (pdf)
Last revised on February 27, 2020 at 16:28:14. See the history of this page for a list of all contributions to it.