ON THE NONUNIQUENESS OF SOLUTIONS OBTAINED WITH SIMPLIFIED VARIATIONAL-PRINCIPLES

The attempt to satisfy subsidiary conditions in boundary value problem s without additional independent unknowns in the form of Lagrange mult ipliers has led to the development of so-called ''simplified variation al principles.'' They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of th e 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 bou ndary value problem. A particularly simple form of the geometrically n onlinear theory of bending of beams is used to the vehicle for this pr oof. The development given in this paper is entirely analytical. Hence , the deficiencies of the investigated simplified variational principl e are fundamental.