CURVATURE AND VARIATIONAL MODELING IN PHYSICS AND BIOPHYSICS
1002(2008); http://dx.doi.org/10.1063/1.2918095View Description Hide Description
These five lectures constitute a tutorial on the Euler elastica and the Kirchhoff elastic rod. We consider the classical variational problem in Euclidean space and its generalization to Riemannian manifolds. We describe both the Lagrangian and the Hamiltonian formulation of the rod, with the goal of examining the (Liouville‐Arnol'd) integrability. We are particularly interested in determining closed (i.e., periodic) solutions.
1002(2008); http://dx.doi.org/10.1063/1.2918094View Description Hide Description
1002(2008); http://dx.doi.org/10.1063/1.2918096View Description Hide Description
The main idea governing this article is to assume that the dynamics associated with a physical model is encoded in the geometry of their so called extended structures. According to this consideration, we collect some geometrical variational models constructed with dynamical variables being curves and surfaces. Consequently, these approaches can be viewed as field theories governed by Lagrangians whose densities are functions of the geometrical extrinsic invariants of their elementary fields. The geometrical models that we exhibit apply to different physical phenomena going from relativistic particles to bosonic string theories. Even to different physical contexts apparently unrelated as sigmamodels and membranes theories, magnetic flows and elastic rods, etc.
1002(2008); http://dx.doi.org/10.1063/1.2918090View Description Hide Description
Liquid crystals are phases of matter with properties intermediate to liquids and crystals. In these lectures, I will consider nematic and smectic liquid crystalline phases which are geometrically frustrated by either chiral molecular interactions, boundary conditions, or background spatial curvature. To resolve this frustration, these lquid crystals allow the introduction of topological defects in their ground state which may then organize into an ordered configuration. In particular, we will consider nematic blue phases, smectic twist‐grain boundary phases, and focal domains in this light.
1002(2008); http://dx.doi.org/10.1063/1.2918091View Description Hide Description
We discuss shapes and shape fluctuations of semiflexible polymers or filaments, which are polymers with an appreciable bending rigidity. The physical properties of semiflexible polymers are governed by their persistence length. On length scales smaller than the persistence length thermal fluctuations can be neglected and polymer shapes are obtained by bending energy minimization. On length scales larger than the persistence length, however, thermal shape fluctuations play an important role and cannot be neglected in general. After a general definition of the persistence length based on the bending rigidity renormalization we will review some problems related to single semiflexible polymers where both variational problems of energy minimization and thermal fluctuations play an important role. We will discuss the buckling instability of semiflexible polymers, their force‐induced desorption, and the shapes of adsorbed semiflexible polymers on structured substrates.
1002(2008); http://dx.doi.org/10.1063/1.2918092View Description Hide Description
This paper presents a collection of experiments in the Calculus of Variations. The implementation of the Gradient Descent algorithm built on cubic‐splines acting as “numerically friendly” elementary functions, give us ways to solve variational problems by constructing the solution. It wins a pragmatic point of view: one gets solutions sometimes as fast as possible, sometimes as close as possible to the true solutions. The balance speed/precision is not always easy to achieve.
Starting from the most well‐known, classic or historical formulation of a variational problem, section 2 describes briefly the bridge between theoretical and computational formulations. The next sections show the results of several kind of experiments; from the most basics, as those about geodesics, to the most complex, as those about vesicles.