On Energy Functions for String-Like Continuous Curves, Discrete Chains, and Space-Filling One Dimensional Structures

The theory of string-like continuous curves and discrete chains have numerous important physical applications. Here we develop a general geometrical approach, to systematically derive Hamiltonian energy functions for these objects. In the case of continuous curves, we demand that the energy function must be invariant under local frame rotations, and it should also transform covariantly under reparametrizations of the curve. This leads us to consider energy functions that are constructed from the conserved quantities in the hierarchy of the integrable nonlinear Schr"odinger equation (NLSE). We point out the existence of a Weyl transformation that we utilize to introduce a dual hierarchy to the standard NLSE hierarchy. We propose that the dual hierarchy is also integrable, and we confirm this to the first non-trivial order. In the discrete case the requirement of reparametrization invariance is void. But the demand of invariance under local frame rotations prevails, and we utilize it to introduce a discrete variant of the Zakharov-Shabat recursion relation. We use this relation to derive frame independent quantities that we propose are the essentially unique and as such natural candidates for constructing energy functions for piecewise linear polygonal chains. We also investigate the discrete version of the Weyl duality transformation. We confirm that in the continuum limit the discrete energy functions go over to their continuum counterparts, including the perfect derivative contributions.

💡 Research Summary

The paper develops a unified geometric framework for constructing Hamiltonian energy functionals for string‑like objects, both in the continuum (smooth curves) and in the discrete setting (piecewise‑linear polygonal chains). The central guiding principle is invariance under local frame rotations; for continuous curves an additional requirement of covariance under re‑parametrizations of the curve is imposed.

In the continuous case the authors start from the Frenet‑Serret description of a space curve (\mathbf{X}(s)) with tangent (\mathbf{t}), normal (\mathbf{n}) and binormal (\mathbf{b}). The curvature (\kappa(s)) and torsion (\tau(s)) are combined into a complex “Hasimoto” field (\psi(s)=\kappa(s)\exp!\big(i\int^{s}\tau,ds’\big)). Because (\psi) transforms as a scalar under local SO(2) rotations of the normal–binormal plane, any functional built solely from (\psi) and its complex conjugate is automatically frame‑rotation invariant. To satisfy re‑parametrization covariance, the authors note that (\psi) carries a weight of one derivative, so integrals of the form (\int ds,\mathcal{F}(\psi,\psi^{*})) are invariant provided (\mathcal{F}) is a scalar density of weight –1.

The key insight is that the infinite hierarchy of conserved quantities of the integrable nonlinear Schrödinger equation (NLSE) provides precisely such scalar densities. The NLSE Lax pair yields an infinite set of Hamiltonians
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