Magnetic chains: From self-buckling to self-assembly

Spherical neodymium-iron-boron magnets are perman-ent magnets that can be assembled into a variety of structures due to their high magnetic strength. A one-dimensional chain of these magnets responds to mechanical loadings in a manner reminiscent of an elastic rod. We investigate the macroscopic mechanical properties of assemblies of ferromagnetic spheres by considering chains, rings, and chiral cylinders of magnets. Based on energy estimates and simple experiments, we introduce an effective magnetic bending stiffness for a chain of magnets and show that, used in conjunction with classic results for elastic rods, it provides excellent estimates for the buckling and vibration dynamics of magnetic chains. We then use this estimate to understand the dynamic self-assembly of a cylinder from an initially straight chain of magnets.Comment: Final version, as publishe

On Density of State of Quantized Willmore Surface-A Way to Quantized Extrinsic String in R^3

Recently I quantized an elastica with Bernoulli-Euler functional in two-dimensional space using the modified KdV hierarchy. In this article, I will quantize a Willmore surface, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, I show that the density of state of the partition function for the quantized Willmore surface is expressed by volume of a subspace of the moduli of the MNV equation.Comment: AMS-Tex Us

Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized Elastica

In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article to those of hyperelliptic curves with general genera. It was proved that the tangential angle of loop soliton is expressed by the Weierstrass hyperelliptic al function for a given hyperelliptic curve $y^2 = f(x)$ with genus $g$.Comment: AMS-Tex, 14 page

Area-Constrained Planar Elastica

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties: not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but, in addition, the embedding itself can be expressed as a local function of the curvature. The configuration space is shown to be essentially one-dimensional, with surprisingly rich structure. Distinct branches of integer-indexed equilibria exhibit self-intersections and bifurcations -- a gallery of plots is provided to highlight these findings. Perturbations connecting equilibria are shown to satisfy a first order ODE which is readily solved. We also obtain analytical expressions for the energy as a function of the area in some limiting regimes.Comment: 23 pages, several figures. Version 2: New title. Changes in the introduction, addition of a new section with conclusions. Figure 14 corrected and one reference added. Version to appear in PR

Statistical Mechanics of Elastica on Plane as a Model of Supercoiled DNA-Origin of the MKdV hierarchy-

In this article, I have investigated statistical mechanics of a non-stretched elastica in two dimensional space using path integral method. In the calculation, the MKdV hierarchy naturally appeared as the equations including the temperature fluctuation.I have classified the moduli of the closed elastica in heat bath and summed the Boltzmann weight with the thermalfluctuation over the moduli. Due to the bilinearity of the energy functional,I have obtained its exact partition function.By investigation of the system,I conjectured that an expectation value at a critical point of this system obeys the Painlev\'e equation of the first kind and its related equations extended by the KdV hierarchy.Furthermore I also commented onthe relation between the MKdV hierarchy and BRS transformationin this system.Comment: AMS-Tex Us

Convergence of Hencky-type discrete beam model to euler inextensible elastica in large deformation: Rigorous proof

The present chapter concerns rigorous homogenization of a Hencky-type discrete beam model, which is useful for the numerical study of complex fibrous systems as pantographic sheets as well as woven fabrics. -convergence of the discrete model towards the inextensible Euler’s beam model is proven and the result is established for placements in Rd in large deformation regime