In chapter 13 of Goldstein's classical mechanics, on page 591 when talking about the Noether's theorem, Goldstein says we need condition 3, which is $$\tag{13.133} \int_{\Omega'}\mathcal{L}(\eta_\rho'(x^\mu),\eta'_{\rho,\nu}(x^\mu),x^\mu)\,dx^4- \int_{\Omega}\mathcal{L}(\eta_\rho(x^\mu),\eta_{\rh...