A general theory of equilibrium paths through critical points. I

Abstract

This paper explores the nonlinear problem of determining the local shape of all possible equilibrium paths through a point where a unique path is not assured. For the sake of rapid insight the treatment is confined to systems having a potential energy depending on any finite number of variables, one direct application being in the theory of structures. The relation with stability is described. The problem is reduced to a readily defined sequence of linear and nonlinear governing equations. These are all expressed explicitly in terms of the orthonormalized eigenvectors of an algebraic eigenvalue problem, and the case of a multiple zero eigenvalue (related to the existence of multiple buckling modes) is thereby concisely treated. Branching points which do not permit changes of load are distinguished from those which do. A comparison with Koiter’s work is given.