nLab pre-symplectic current

Contents

Idea

A local Lagrangian density $L \in \Omega^{p+1,0}(E)$ on a jet bundle $J^\infty(E)$ of some field bundle $E \to \Sigma$ induces a differential form

\theta \in \Omega^{p,1}(E)

unique up to terms in the image of the horizontal derivative, given by

d L = \delta_{EL} L - d_H \theta \,,

where $\delta_{EL}$ is the Euler-Lagrange operator. Its image under the vertical derivative

\omega \coloneqq d_V \theta \in \Omega^{p,2}(E)

is called the pre-symplectic current. Accordingly $\theta$ itself is also called the presymplectic potential current.

This is because for any choice of $p$ -dimensional submanifold $\Sigma_p \hookrightarrow \Sigma$ , the transgression of variational differential forms of $\omega$ to $\Sigma_p$ is a presymplectic form on the on-shell space of sections on the infinitesimal neighbourhood of $\Sigma_p$ . This is the covariant phase space.

Last revised on December 29, 2023 at 13:18:36. See the history of this page for a list of all contributions to it.