Extension of the classical Cartan form

Abstract

For a given Lagrangian $L$ a differential form ${\ensuremath{\theta}}_{L}^{l}$ is introduced which generalizes the classical notion of the Cartan form ${\ensuremath{\theta}}_{L}^{l}$. This extended Cartan form ${\ensuremath{\theta}}_{L}^{l}$ still gives rise to the same field equations (Euler-Lagrange equations), but has the added feature that $d{\ensuremath{\theta}}_{L}^{l}=0$ if and only if the Euler-Lagrange equations for $L$ vanish identically. This has the important consequence that the symmetries of a given Lagrangian $L$ are precisely those diffeomorphisms which transform $d{\ensuremath{\theta}}_{L}^{l}$ into itself, a property not shared by the classical form $d{\ensuremath{\theta}}_{L}^{l}$.