nLab
Euler-Lagrange complex

Contents

Idea

Given a smooth bundle EΣE \to \Sigma over a smooth manifold Σ\Sigma, then its Euler-Lagrange complex is a resolution of the constant sheaf of locally constant functions on the jet bundle J EJ^\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 ndim(Σ)n \coloneqq dim(\Sigma) the dimension of Σ\Sigma, the following differential is

  1. the Euler-Lagrange operator δ El\delta_{El}

  2. followed by the Helmholtz operator δ Heml\delta_{Heml}

0Ω 0,0(J E)d HΩ 1,0(J E)d Hd HΩ n,0(J E)δ EL 1(J E)δ Helm 2(J E)δ V 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{\delta_{EL}}{\to} \mathcal{F}^1(J^\infty E) \stackrel{\delta_{Helm}}{\to} \mathcal{F}^2(J^\infty E) \stackrel{\delta_V}{\to} \cdots

Hence the elements in the Euler-Lagrange complex have the following interpretation

Properties

Proposition

The cochain cohomology of the Euler-Lagrange complex

0Ω 0,0(J E)d HΩ 1,0(J E)d Hd HΩ n,0(J E)E 1(J E)δ V 2(J E)δ V 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

is isomorphic to the de Rham cohomology of the total space EE of the given fiber bundle.

(Anderson 89, theorem 5.9).

References

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)

Revised on December 5, 2017 05:05:16 by Urs Schreiber (46.183.103.8)