nLab Euler-Lagrange form

Definition

Given a smooth bundle $E \to \Sigma$ with $\Sigma$ a smooth manifold of dimension $n$, then a horizontal $n$-form $\mathbf{L}$ on the jet bundle $Jet(E)$ is a local Lagrangian. Its de Rham differential has a unique decomposition into a source form $\mathbf{E}$ and a horizontally exact form (with respect to the variational bicomplex)

$$d \mathbf{L} = \mathbf{E} - d_H \theta \,.$$

This source form $\mathbf{E}$ is the Euler-Lagrange form of $\mathbf{L}$. It vanishes precisely at those points which are solutions to the Euler-Lagrange equations induced by $\mathbf{L}$.

The combination $\rho \coloneqq \mathbf{L} + \theta$ is the corresponding Lepage form.

References

• G. J. Zuckerman, Action principles and global geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (pdf)