# Confusion about Noether's theorem

In my field theory class we recently derived Noether's theorem: We consider a infinitessimal transformation $$\phi \to \phi + \epsilon \,\delta\phi$$ of our field which preserves action i. e. $$\delta S = 0$$. This last condition is supposedly equivalent to $$\delta \mathcal{L}$$ being a divergence, i. e. $$\delta \mathcal{L} = \epsilon\,\partial_\mu I^\mu$$. Then you can expand \begin{align} \delta \mathcal{L} &= \frac{\partial \mathcal{L}}{\partial \phi} \epsilon \, \delta \phi + \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \epsilon \partial_\mu(\delta\phi) \\ &= \left( - \partial_\mu \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} + \frac{\partial \mathcal{L}}{\partial \phi}\right) \epsilon \, \delta \phi + \partial_\mu \left( \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi\right) \epsilon \end{align}

The first term vanishes by the eqns. of motion and so we get $$\partial_\mu \left( \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi - I^\mu\right) = 0$$ i.e. $$\frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi - I^\mu$$ is a conserved current.

Two things confuse me here:

We didn't use that the transformation is supposed to be a symmetry of the system (that is $$\delta S = 0$$). In a point-particle setting (that is a field that only depends on time) we can have $$L = T - V$$ with $$\partial V / \partial q \neq 0$$, but we can look at a translation $$q \to q + \epsilon$$ which gives us $$L \to L + \epsilon \frac{\partial}{\partial t} \left( - t \frac{\partial V}{\partial q} \right)$$ so $$L$$ is only changed up to a "divergence" and the resulting conserved quantity is $$p + t \frac{\partial V}{\partial q}$$, which is indeed conserved, even though our system was not space-homogeneous.

The other source of confusion (which I guess is related to the first) is this argument that $$\delta S = 0$$ is equivalent to $$\delta \mathcal{L} = \epsilon \partial_\mu I^\mu$$. In $$\mathbb{R}^n$$, every scalar function can be written as a divergence, so this doesn't seem to add up. Is $$I^\mu$$ maybe supposed to be a function of the fields only?