The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present a Hamiltonian framework for the binormal equation in higher-dimensions and its explicit solutions that collapse in finite time. More generally, we also describe the geometry behind Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. In particular, the Madelung transform between the Schrödinger-type equations on wave functions and Newton's equations on densities turns out to be a Kähler map between the corresponding phase spaces. This is a joint work with C.Yang, G.Misiolek and K.Modin.