Contents

Idea

Given a smooth bundle over a smooth manifold , then its Euler-Lagrange complex is a resolution of the constant sheaf of locally constant functions on the jet bundle 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 the dimension of , the following differential is

1. the Euler-Lagrange operator

2. followed by the Helmholtz operator

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

• in degree : local Lagrangians;

• in degree : Euler-Lagrange equations of motion;

• in degree : trivial local Lagrangians;

• in degree : obstructions for equations of motion to be variational, i.e. to be the Euler-Lagrange equations of a local Lagrangian.

Properties

(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)