# Noether's theorem under arbitrary coordinate transformation

Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

Suppose our action is of the form $$S = \int d^4x\, \mathcal{L}(\phi,\partial_\mu\phi).\tag{1}$$

if $$x \rightarrow x'$$ then if $$S \rightarrow S'$$ where
$$S' = \int d^4x'\, \mathcal{L'}(\phi',\partial_\mu\phi').\tag{2}$$

But from calculus we know that $$S=S'$$ so does that mean that every change of variable correspond to a conserved quantity? why the quantities conserved under Poincare transformation, for example, is more especial?

The action shown in the question is a functional of $$\phi$$, not of $$x$$. A change of the integration variable $$x$$ is just a relabeling of the index set. It does not transform the dynamic variables $$\phi$$ at all, so no: a change of variable does not correspond to a conserved quantity.
More explicitly, if $$y(x)$$ is a monotonic smooth function of $$x$$, then $$\int d^4y\ {\cal L}\left(\phi\big(y(x)\big),\, \frac{\partial}{\partial y_\mu}\phi\big(y(x)\big)\right) = \int d^4x\ {\cal L}\left(\phi(x),\frac{\partial}{\partial x_\mu}\phi(x)\right) \tag{1}$$ identically, for any $${\cal L}$$ whatsoever (as long as it depends on $$x$$ only via $$\phi$$). This is just a change of variable (a relabeling of the index-set), and there is no associated conserved quantity.
In contrast, suppose that the action has this property: $$\int d^4x\ {\cal L}\left(\phi\big(y(x)\big),\, \frac{\partial}{\partial x_\mu}\phi\big(y(x)\big)\right) = \int d^4x\ {\cal L}\left(\phi(x),\frac{\partial}{\partial x_\mu}\phi(x)\right). \tag{2}$$ Unlike equation (1), equation (2) is not identically true for any $${\cal L}$$ and any $$y(x)$$, though it may be true for some choices of $${\cal L}$$ and $$y(x)$$. The transformation represented in equation (2) replaces the original function $$x$$, namely $$\phi(x)$$, with a new function of $$x$$, namely $$\phi\big(y(x)\big)$$. This is the kind of transformation we have in mind when we talk about Poincaré invariance and its associated conserved quantities: it is a change of the function $$\phi$$ which we then insert into the original action, not a change of the integration variable.
2. In particular, if we consider a passive coordinate transformation of the system, then the action $$S$$ is trivially invariant, and we can e.g. use the trick with $$x$$-dependent infinitesimal $$\epsilon(x)$$ to conclude that the corresponding full Noether current $$J^{\mu}$$ vanishes identically.