The Navier-Stokes equations can be geometrized in the following form:
$$\dot{u} + \nabla_u u =\nu \Delta (u)+ df^* $$
$$d^* u=0$$
$\nabla_X Y$ is the connection $dY(X)$. If we define $u=\dot{\gamma}$, we recognize the equation of geodesics:
$$ \nabla_{\dot{\gamma}}\dot{\gamma}=0$$
Can we solve the Navier-Stokes equations with help of a lagrangian formalism?