nLab source form

Definition

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):

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

In variational calculus

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.

Related concepts

• variational bicomplex

• evolutionary vector field, evolutionary derivative

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)