It is a very well-known fact that any conservation law associated with some given PDE has an associated invariance (by Noether's Theorem). However, it is completely mysterious for me how to compute/derive these conservation laws just by knowing the invariances of the equation. For example, the one-dimensional nonlinear wave equation $$ u_{tt}-u_{xx}+f(u)=0, \qquad (t,x)\in\mathbb{R}\times\mathbb{R}, $$ is invariant under space translations. On the other hand, it is "well-known" that associated to this space translation invariance is the momentum conservation of the equation, that is, $$ P(u,v)(t):=\int_{\mathbb{R}} u_x(t,x)v(t,x)dx=\int_{\mathbb{R}}u_{0,x}(x)v_0(x)dx=P(u,v)(0). $$ Nevertheless, I have no idea how to derive this conservation law (generally speaking) just by knowing that the equation is invariant under space translations. What about time-translations for example, what is its associated conservation law? Please, don't misunderstand me, I do know how to explicitly derive the momentum conservation directly from the equation, what I would like to know is how to derive it from the space-translations invariance. Any hint suggested reading or answer is very welcome!