Given a smooth bundle $E \to \Sigma$, then a differential form on its jet bundle $J^\infty_\Sigma(E)$ is called a *source form* if it is of vertical degree 1 (with respect to the variational bicomplex) and its evaluation on a vector field depends only on the projection of that vector field to a vector field on $E$ itself (e.g. Zuckerman 87, p. 6):

$\Omega^{k,1}_{\Sigma,source}(E)
\;\coloneqq\;
\Omega^{k,0}_\Sigma(E) \wedge \delta C^\infty(E)
\,.$

Given a local Lagrangian $\mathbf{L}$, i.e. a horizontal form on $Jet(E)$ of maximal horizontal degree, then there is a unique source form $\mathbf{E}$ such that

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

for some form $\theta$. This $\mathbf{E}$ is the Euler-Lagrange form of $\mathbf{L}$.

The sum

$\rho \coloneqq \mathbf{L} + \theta$

is the corresponding Lepage form.

- Gregg J. Zuckerman,
*Action principles and global geometry*, in: Shing-Tung Yau (ed.)*Mathematical Aspects of String Theory*, World Scientific (1987) 259-284 [pdf, doi:10.1142/0383]

Last revised on September 14, 2023 at 17:59:08. See the history of this page for a list of all contributions to it.