  In this work we derive Noether Theorems for energies of the form
\begin{equation*} E(u)=\int_\Omega
L\left(x,u(x),(-\Delta)^\frac{1}{4}u(x)\right)dx \end{equation*} for
Lagrangians exhibiting invariance under a group of transformations acting
either on the target or on the domain of the admissible functions $u$, in terms
of fractional gradients and fractional divergences. Here $\Omega$ stays either
for an Euclidean space $\mathbb{R}^n$ or for the circle $\mathbb{S}^1$. We then
discuss some applications of these results and related techniques to the study
of nonlocal geometric equations and to the study of stationary points of the
half Dirichlet energy on $\mathbb{S}^1$. In particular we introduce the
$\frac{1}{2}$-fractional Hopf differential as a simple tool to characterize
stationary point of the half Dirichlet energy in
$H^\frac{1}{2}(\mathbb{S}^1,\mathbb{R}^m)$ and study their properties. Finally
we show how the invariance properties of the half Dirichlet energy on
$\mathbb{R}$ can be used to obtain Pohozaev identities.
