  In 2000 Constantin showed that the incompressible Euler equations can be
written in an "Eulerian-Lagrangian" form which involves the back-to-labels map
(the inverse of the trajectory map for each fixed time). In the same paper a
local existence result is proved in certain H\"older spaces $C^{1,\mu}$.
  We review the Eulerian-Lagrangian formulation of the equations and prove that
given initial data in $H^s$ for $n\geq2$ and $s>\frac{n}{2}+1$, a unique
local-in-time solution exists on the $n$-torus that is continuous into $H^s$
and $C^1$ into $H^{s-1}$. These solutions automatically have $C^1$
trajectories.
  The proof here is direct and does not appeal to results already known about
the classical formulation. Moreover, these solutions are regular enough that
the classical and Eulerian-Lagrangian formulations are equivalent, therefore
what we present amounts to an alternative approach to some of the standard
theory.
