H.A. Mang, G. Hofstetter, R.H. Gallagher:
"On an Alleged Philosopher's Stone Concerning Variational Principles with Side Conditions";
International Journal for Numerical Methods in Engineering, 21 (1985), S. 1971 - 1999.
In general, the Lagrange multiplier method (LMM) is used to incorporate subsidiary conditions into variational principles. The Lagrange multipliers represent an additional field of independent variables. The attempt to satisfy subsidiary conditions without employing additional independent unknowns has led to the development of simplified variational principles (SVP). They are characterized by expressing Lagrange multipliers in terms of orginial field variables by means of the Euler-Lagrange equations for the multipliers, providing their physical interpretation.
In the first part of the theoretical investigation, systems with infinitely many degrees-of-freedom are studied. It is shown that the Euler-Lagrange equations of a LMM based on a modification of the principle of minimum of potential energy (PMIPE) do not hold unconditionally, that is, for arbitrary subsidiary conditions, for the corresponding SVP. The second part of the theoretical investigation is concerned with systems with a finite number of degrees-of-freedom. The finite element method (FEM) is employed to discuss and compare the characteristics of the LMM and of the corresponding SVP. Contrary to the former, the latter are found to be problem-dependent. Several shortcomings of the SVP are listed, including the possibility of obtaining an infinite sequence of singular coefficient matrics in the process of a systematic mesh refinement. Consequently, convergence of finite element solutions to the true solution in the limit of finite element representations is not guaranteed. The theoretical findings are corroborated by the results of a detailed numerical study.