Problem of the identical vanishing of Euler-Lagrange derivatives in field theory

Abstract

I prove that the necessary and sufficient condition for two Lagrangian densities ${\mathcal{L}}_{1}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}})$ and ${\mathcal{L}}_{2}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}})$ to have exactly the same Euler-Lagrange derivatives is that their difference $\ensuremath{\Delta}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}})$ be the divergence of ${\ensuremath{\omega}}^{\ensuremath{\mu}}({\ensuremath{\psi}}^{A};{{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}};{x}^{\ensuremath{\mu}})$ with a given dependence on ${{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}}$. The main point is that ${\ensuremath{\omega}}^{\ensuremath{\mu}}$ depends on ${{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}}$ but $\ensuremath{\Delta}$ does not depend on second derivatives of the field ${\ensuremath{\psi}}^{A}$. Therefore, the function $\ensuremath{\Delta}$ need not be linear in ${{\ensuremath{\psi}}^{A}}_{,\ensuremath{\alpha}}$.