论拉氏乘子法及其唯一性问题

本文指出文13](1985)对于拉氏乘子法的最近论点仍旧是先验的,并不是国际上大家所公认而又证实了的“古老的数学概念”(1983),该文所赖以立论的三个实例,都不成立。所说明的,不是象文中所称的那样,“在力学问题中正确应用拉氏乘子法的要点”,恰好相反,文13]很不正确地应用了拉氏乘子法,从而达到了错误结论,甚至只能求助于所谓“猜谜语“的方法。 本文也指出拉氏乘于是可以根据拉氏乘子法唯一地识別的,文10]、文16]说拉氏乘子的不唯一性应是对拉氏乘子法的误解所引起的。 本文讨论的弹性力学问题是非线性弹性体的一般弹性力学问题,其应力应变关系是非线性的,当应变很小可以略去其非线性项时,其结果可以还原为线性弹性体的各种广义变分原理。因此,不论Hellinger-Reissner 原理或胡-鹫原理都是本文所讨论的非线性弹性体的广义变分原理的近似特例。

ON LAGRANGE MULTIPLIER AND ITS UNIQUENESS PROBLEM

This paper shows that Hu Hai-chang's recent point of view (1985) on the method of Lagrange multiplier is of an apriori character, and is hot the traditional mathematical concept that have been known and used in most of the international circles (1983) Hu's concepts are based upon three practical examples, however, all of them are not valid. He tried to show the important point;, that have been used for the correct application of the method of Lagrange multiplier in the mechanics. However, it is not worked as what he considered; on the other hand, Hu actually did used incorrectly the method of Lagrange multiplier, and this arrived at wrong conclusion. In the lasr, he had to resort to the method of puzzle guessing. in order to avoid the mounted difficulties.This paper also shows that the Lagrange multipliers can be uniquely identified by the method of Lagrange multipliers. It is also shown that there are misunderstandings concerning the method of Lagrange multiplier in Chang-Xin Wong (1981) and Hu-Hia Hu Chang (1983) works, where they discovered the so-called non-unqueness in the identification of Lagrange multiplier.In this paper, the elasticity problems are studied in general on the bases of non-linear elasticity, in which the stress-strain relation is nonlinear. When the strain is small so that the nonlinear terms can be neglected, the results can be reduced into various generalized varia-tional principles for linear elastic problems. Therefore, no matter Hellinger-Reissner principle or Hu-Washizu principles, all of them are but the specialized approximation of the general non-linear elastic problems discussed in this paper.