Legendre变换的机械化求法

把Lagrange方程和Hamilton方程组写成矩阵形式,其系数矩阵的元素是非线性的微分算子．参考矩阵多元多项式的带余除法的方法把微分算子的矩阵转化成以多项式为元素的矩阵,应用待定系数法通过Mathematica机械性计算得到了含有可变参数的一类变换,其中当参数取零时,它就是Legendre变换．经证明,这一类变换当参数不为零时,应用该变换,也可把n=1时的Lagrange方程导向Hamilton正则方程组,并初步讨论了n维的情况．这是数学机械化的方法在分析力学的非线性理论问题中的一次具体实现。

Mathematical Mechanization Method on the Legrendre Transformation

The Hamilton equations and Lagrange equation are written on Matrix forms,in which the elements of coefficient matrix are non linear differential operators.Based on pseudo division algorithm for matrix multivariable polynomial,the matrix of differential operators is changed into polynomial matrix.By using an indeterminate coefficient method,a sorts of transforms including a variable parameter is generated by Mathematica,and if the variable parameter is equal to zero,it is the Legendre transformation.It is proved that the Lagrange equation lead to the Hamilton equations by using these transformations when the variable parameter is not equal to zero and the condition of n dimension is discussed.It is a concrete realization in non linear theory problem of analytical mechanics by mathematical mechanization.