Two-dimensional Laplacian growth can be mapped onto Hamiltonian dynamics

TL;DR

This paper demonstrates that the growth of a 2D surface in a Laplacian field can be described using Hamiltonian dynamics through conformal mapping and singularity analysis, providing a new theoretical framework.

Contribution

It introduces a novel mapping of 2D Laplacian growth onto Hamiltonian dynamics, including explicit conditions and an example of an integrable case.

Findings

Mapping of surface growth dynamics to Hamiltonian form

Explicit condition for the transformation's existence

Solution of a separable, integrable Hamiltonian case

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.