Hamiltonian vector fields on a space of curves on the 3-sphere

Abstract

This thesis reviews aspects related to the integrability of a Hamiltonian system on a space of arc length parametrized curves on the unit sphere $S^3$ in $\mathbb{R}^4$ of a fixed length $L$. In particular, we find that the flow of the Hamiltonian vector field corresponding to the total torsion function $X(s) \mapsto \displaystyle\int_o^{L} \tau(s) ds$ generates the curve shortening equation. Additionally, we show that the total torsion function belongs to a hierarchy of Poisson commuting functions.