The new theory
This chapter is devoted to a new equilibrium theory of liquid crystals, which has been formulated by J. L. Ericksen in a rigorous way, though it had been anticipated in some earlier works of De Gennes, C. P. Fan and M. J. Stephen. It focuses on special issues such as the space dimension of defects in the minimizes of F. Frank’s energy functional. The chapter shows that when the one-constant approximation to Frank’s energy applies there is indeed only one point defect which is essentially the prototype of all others, in that it generates all singularities admissible in a minimizer of the energy functional. It explains how plane and line defects do actually arise in the solutions of specific variational problems posed within the new theory. The Hamilton-Jacobi method is usually referred to as the dynamic programming method in control theory.