Energy functions for stringlike continuous curves, discrete chains, and space-filling one dimensional structures

Abstract

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 Schrödinger 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. * Electronic address: hushuangwei@gmail.com † Electronic address: yjiang@shu.edu.cn ‡ Electronic address: Antti.Niemi@physics.uu.se 1 arXiv:1210.7371v1 [hep-th]

Long range correlations and folding angle with applications to α-helical proteins

“…It describes collapsed chiral homopolymers as local energy minima in terms of the backbone bond and torsion angles. The detailed derivation of (8) can be found in [29]. Here it suffices to state that the energy function (8) can be shown to be a long-distance limit that describes the full microscopic energy of a folded protein [20].…”

Gauge fields, strings, solitons, anomalies, and the speed of life

“…For a detailed discussion of (65) we refer to [9,10]. Since the arguments that lead to (65) are based entirely on general symmetry principles that are universally valid, the result (65) is unique for small fluctuations around a given background: The energy (65) engages the complete set of gauge invariant quantities, in terms of the bond and torsion angles, that emerge at leading order in a systematic Taylor expansion of the full energy around its local extremum.…”

“…From the extrinsic string geometry we can construct an exemplar Abelian Higgs Multiplet [8][9][10]. For this we observe that (8) does not engage the normal and bi-normal vectors.…”

“…In addition we have included the Proca mass term (the last term) [9,10]. Even though the Proca mass does not appear in the integrable NLS hierarchy, it does have a claim of gauge invariance and as such it is often accounted for.…”

“…the unit length bi-normal vector b = x s × x ss ||x s × x ss || (9) and the unit length normal vector, n = b × t…”