On Griffiths' formalism of the calculus of variations
Abstract of thesis entitled
OF THE CALCULUS OF VARIATIONS
submitted by CHAN Ka-bo
for the Degree of Master of Philosophy
at The University of Hong Kong
in August 2004
Associated to various physical or geometric objects there are certain important quantities. For example, associated to a closed curve in the plane, there is the area of the region it encloses; and associated to an elastic wire in space, there is the energy stored in it. It is important to determine the position of these objects for which these quantities achieve their extremum. Such problems are in general known as ?alculus of Variations Problems? Mathematically, this accounts to determining an unknown function y = y(x) that maximizes or minimizes functional of the form
F(y)= F(x;yy>)dx, for j = 0,..., k, i=1,...,n,
where y = (y1,... ,yn), yf = g, y% are smooth functions for all i, and F is a smooth function, while at the same time satisfying certain side conditions.
The classical formalism of the Calculus of Variations works well in many physical problems but it has serious drawbacks when applied to intrinsic geometric problems. Take as an example
F(?)= / L(?1(s),...,?n-1(s))ds,
where ? is a curve in euclidean n-space parametrized by arc length s, ? 1(s), ..., ?n-1(s) are the n - 1 curvatures associated to ? which are well-known
to be euclidean invariants, and L is a smooth function. For instance, if L = 1, F is just the arc length of ?; if L = f, F is the energy stored in the curve ?. For such a simple problem the classical formalism leads to complicated computations and provides only very partial and unsatisfactory results. Due to this drawback, by utilizing the theories and techniques of exterior differential systems, P. Griffiths has developed a new approach of solving general variational problems which provides an effective computational tool for intrinsic geometric variational problems.
In this thesis, basic terminologies and preliminaries of the theory of exterior differential systems are illustrated in Chapter 1. In Chapter 2, Griffiths?formalism on the calculus of variations via exterior differential systems is discussed. Chapter 3 is devoted to applications of the formalism. These include a detailed treament of the classical isoperimetric problem and the computation of all C?-invariants on the space of embedded loops on a general surface S.
School:The University of Hong Kong
School Location:China - Hong Kong SAR
Source Type:Master's Thesis
Keywords:calculus of variations
Date of Publication:01/01/2005