Two-dimensional Laplacian growth can be mapped onto Hamiltonian dynamics

Abstract

It is shown that the dynamics of the growth of a two dimensional surface in a Laplacian field can be mapped onto Hamiltonian dynamics. The mapping is carried out in two stages: first the surface is conformally mapped onto the unit circle, generating a set of singularities. Then the dynamics of these singularities are transformed to Hamiltonian action-angle variables. An explicit condition is given for the existence of the transformation. This formalism is illustrated by solving explicitly for a particular case where the result is a separable and integrable Hamiltonian.