Hamiltonian motions of plane curves and formation of singularities and bubbles

TL;DR

This paper introduces Hamiltonian deformations of plane curves, demonstrating how these deformations can lead to singularities, bubbles, and genus changes, modeled by hydrodynamical type equations.

Contribution

It defines a new class of Hamiltonian deformations for plane curves and analyzes their role in singularity and bubble formation.

Findings

Hamiltonian deformations can produce cusps and nodes.

Solutions describe genus changes in curves.

Hydrodynamical equations model the deformation processes.

Abstract

A class of Hamiltonian deformations of plane curves is defined and studied. Hamiltonian deformations of conics and cubics are considered as illustrative examples. These deformations are described by systems of hydrodynamical type equations. It is shown that solutions of these systems describe processes of formation of singularities (cusps, nodes), bubbles, and change of genus of a curve.