On the Nonuniqueness of Solutions Obtained With Simplified Variational Principles

Abstract

The attempt to satisfy subsidiary conditions in boundary value problems without additional independent unknowns in the form of Lagrange multipliers has led to the development of so-called “simplified variational principles.” They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of the original variables. It is shown that the conversion of a variational principle with subsidiary conditions to such a simplified variational principle may lead to the loss of uniqueness of the solution of a boundary value problem. A particularly simple form of the geometrically nonlinear theory of bending of beams is used as the vehicle for this proof. The development given in this paper is entirely analytical. Hence, the deficiencies of the investigated simplified variational principle are fundamental.