II - Degenerate Duality, Catastrophes and Saddle Functionals

Abstract

This chapter describes the degenerate duality, catastrophes, and saddle functionals. The statement of a single stationary principle may appear in different but equivalent forms. It is a preliminary study of the transformations that may be needed to pass between equivalent variational principles. The ideas are fixed by concentrating on the finite-dimensional case so that one can make a precise connection with the critical point theory of Morse functions, and this approach also allows relating objective directly with the classification of degenerate stationary points, which is a concern of elementary catastrophe theory. Saddle functions appear as one of the forms from which stationary principles may be generated. A diffeomorphism can be found to express any Morse function of two variables, quadratic or not, exactly as one of the three standard forms near a stationary point. There is no smooth coordinate transformation that can convert one of the three standard Morse functions into either of the other two.