nLab Euler-Lagrange complex

Idea

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

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

2. followed by the Helmholtz operator $\delta_{Helm}$

$$0 \to \mathbb{R} \to \Omega^{0,0}(J^\infty E) \stackrel{d_H}{\to} \Omega^{1,0}(J^\infty E) \stackrel{d_H}{\to} \cdots \stackrel{d_H}{\to} \Omega^{n,0}(J^\infty E) \stackrel{\delta_{EL}}{\to} \mathcal{F}^1(J^\infty E) \stackrel{\delta_{Helm}}{\to} \mathcal{F}^2(J^\infty E) \stackrel{\delta_V}{\to} \cdots$$

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

• 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.

Properties

Proposition

The cochain cohomology of the Euler-Lagrange complex

$$0 \to \mathbb{R} \to \Omega^{0,0}(J^\infty E) \stackrel{d_H}{\to} \Omega^{1,0}(J^\infty E) \stackrel{d_H}{\to} \cdots \stackrel{d_H}{\to} \Omega^{n,0}(J^\infty E) \stackrel{E}{\to} \mathcal{F}^1(J^\infty E) \stackrel{\delta_V}{\to} \mathcal{F}^2(J^\infty E) \stackrel{\delta_V}{\to} \cdots$$

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

(Anderson 89, theorem 5.9).

Related concepts

• variational sequence

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)