Optimal Functions and Nanomechanics III

Many problems in the sciences and engineering seek to find a shape or function that minimises or maximises some quantity. For example, an engineer may design a yacht's hull to minimise drag. And in nature, the shape that a complicated protein might adopt is determined in part by the lowest-energy state available to the protein during the folding process. The Calculus of Variations extends familiar calculus techniques to answer questions regarding optimal geometry or functions. The Calculus of Variations is applicable to almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics and string theory. In this course we will consider, in particular, problems from Nanoscience. Nanoscience is a multidisciplinary field at the nexus of physics, chemistry and engineering. Materials and systems that may be very well understood at the macroscale can often exhibit surprising phenomena at the nanoscale. Topics covered are: Classical Calculus of Variations problems such as the geodesic, catenary and brachistochrone; derivation and use of the Euler-Lagrange equations; multiple dependent variables (Hamilton's equations) and multiple independent variables (minimal surfaces); constrained problems, problems with variable end points and those with non-integral constraints; conservation laws and Noether's theorem; computational solutions using Euler's finite difference and Rayleigh-Ritz methods. Many of the examples considered will draw from continuum modelling of the intermolecular interaction potential utilizing special functions (such as gamma, beta, hypergeometric and generalized hypergeometric functions of two variables) and by application of Euler's elastica.