Affine Elastic Curves in R^2 and R^3

This book is devoted to the study of the variational properties of affine curves invariant under centroaffine transformations or equiaffine transformations in two and three-dimensional affine space. It can be considered as a counterpart of the study of the classical Euclidean elastic curves. The basic idea here is to replace the Euclidean space with an affine space. We try to develop some properties about the affine curve which is invariant under some specific subgroups of the affine groups A(2) and A(3). We consider the critial curves of the curvature energy functionals related to the affine curvature or centroaffine curvature of the nondegenerate curve or starlike curve repectively. We completely solve the motion equation by using the Jacobi elliptic functions and finally solve the structure equation by using Killing fields and the classification of the special linear group for the starlike curves.