  Theoretical developments in the field of Lagrangian turbulence are made
difficult by the fact that equations governing the evolution of lagrangian
flows are implicit contrary to eulerian flows. In this article, an {\it exact}
explicit equation for incompressible lagrangian fluid flows at high-Reynolds
number is constructed. The method to arrive at the equation of motion and the
proof that it describes the motion of an incompressible fluid are provided. A
truncated version of this new equation is used to show how the lagrangian
turbulent spectrum ($E_\mathrm{lag}(\omega)$) could be inferred. This exercise
showed a complex interrelation between the stirring force field and the flow
itself in the lagrangian turbulence framework whereas the stirring is not
affected by the flow in the eulerian point of view. The result is that
$E_\mathrm{lag}(\omega)\propto \varepsilon\,\omega^{-2}$ seems independent upon
the way the fluid is stirred in the inertial range for a given dissipated power
$\varepsilon$. It also showed that $E_\mathrm{lag}(\omega)\sim \omega^{-s}$
with $0\leq s\leq 1/2$ (depending on the stirring) at low-$\omega$ and
$\sim\omega^{-4}$ in the viscous range at high-$\omega$.
