= Noether current for a constant shift with second derivatives
{c}
{title2=$j^\mu=\mathcal L_{\phi_{,\mu}}-\partial_\nu\mathcal L_{\phi_{,\mu\nu}}$}
If a <Lagrangian density> depending on first and second derivatives has no undifferentiated field, the <scalar-field shift symmetry> $\delta\phi=\epsilon$ gives the displayed <Noether current>. Its <divergence> is minus the <Euler-Lagrange field equation with second derivatives>, so it is conserved <on shell>. The spatial integral of $j^0$ is constant only if the boundary current vanishes. For $\mathcal L=\rho\phi_t^2/2+\alpha\phi_x^2/2-\beta\phi_{xx}^2/2$, it gives $j^0=\rho\phi_t$, $j^x=\alpha\phi_x+\beta\phi_{xxx}$. Fixed transverse endpoint positions break the global shift symmetry of the boundary problem; support forces can then change the total transverse momentum even though the bulk continuity equation still holds.
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