偏微分方程组的对称群及其在弹性力学方程组中应用

Symmetry of PDEs and their applications to elasticity equations

给出了非退化线性偏微分方程组及二次型泛函对称群的不变向量场的一般形式和一类特殊形式非线性偏微分方程组对称群的简化计算条件；利用以上结论及作者以往工作，借助符号运算语言ＭａｔｈｅｍａｔｉｃａＴＭ计算了平面弹性力学方程组一阶Ｌｉｅ－Ｂａｃｔｌｕｎｄ对称群的不变向量场，以及应力函数对应的三维弹性力学方程组的Ｌｉｅ代数．为构造弹性力学方程组的一类广泛精确解及守恒律提供了必要的基础。

The general form of symmetry of nondegenevate linear partial differential equations (PDEs) and quadratic functional and simplified condition of calculations of symmetry for a special class of non linear PDEs are given. Using the above results and the work previously done by authors and with the help of symbolic calculation language Mathematica TM , the authors calculate the first order Lie Bcklund invariant vector fields of high symmetry for plan elasticity equations and the Lie algebra of the stress function corresponding equations in 3D elastics, which will give a necessary basis for constructing a widespread exact solutions and conservation laws to elasticity equations. The calculations also illustrate that the results above decline significantly the work load of calculation of symmetry for a class of PDEs.