The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics

Abstract

The paper defines a symplectic form on an infinite dimensional Fréchet manifold of framed curves over the three dimensional space forms. The curves over which the symplectic form is defined are called horizontal-Darboux curves. It is then shown that the projection on the Lie algebra of the Hamiltonian vector field associated with the functional $f=\frac{1}{2}{\int }_0^L{\kappa }^2(s)ds$ satisfies Heisenberg's magnetic equation (HME), $\frac{\partial \Lambda }{\partial t}(s,t)=\frac{1}{i}[\Lambda (s),\frac{{\partial }^2\Lambda }{\partial s^2}(s,t)]$ in the space of Hermitian matrices for the hyperbolic and the Euclidean case, and $\frac{\partial \Lambda }{\partial t}(s,t)=[\Lambda (s),\frac{{\partial }^2\Lambda }{\partial s^2}(s,t)]$ in the space of skew-Hermitian matrices for the spherical case. It is then shown that the horizontal-Darboux curves are parametrized by curves in $S{U}_2$, which along the solutions of (HME) satisfy Schroedinger's non-linear equation (NSL)

It is also shown that the critical points of $\frac{1}{2}{\int }_0^L{\kappa }^2(s)ds$, known as the elastic curves, correspond to the soliton solutions of (NSL). Finally the paper shows that the modifed Korteweg–de Vries equation or the curve shortening equation are Hamiltonian equations generated by $f_1={\int }_0^L{\kappa }^2(s)\tau (s)ds$ and $f_2={\int }_0^L\tau (s)ds$ and that $f_0=\frac{1}{2}{\int }_0^L{\kappa }^2(s)ds$, $f_1$ and $f_2$ are all in involution with each other.