拉格朗日早年对其变分方法的参数化与发展

Lagrange's parameterization and generalizations of his method of variations in his earlier researches

拉格朗日"变分方法"(亦称δ-算法)的引进,堪为变分法早期发展中的一次变革。然而在最初提出这一新方法时,拉格朗日却经历了由非参数向参数表示形式的转变,实现了该方法的参数化。依据原始文献,首先解析了拉格朗日δ-算法的非参数表示和参数表示,然后探讨了在参数化过程中他对变分法相关理论及应用所作的革新与发展。研究表明:通过参数化改造,拉格朗日对变分问题、变分方程、横截性条件以及变分法的力学应用——最小作用原理等均做出了拓广或发展。拉格朗日对其变分方法的参数化不仅开阔了变分法研究的范围,而且赋予了变分法在力学应用中的重大价值。

The invention of Lagrange's method of variations,which is now also known as δ-algorithm,ushered in a new epoch in the earlie history of the calculus of variations. During the introduction of the new method in his earlier researches,Lagrange experienced the change in his approach from the non-parametric to the parametric representation. Based on some original literature,the parameterization of his method of variation is thoroughly documented,and the innovations and development of the theory of the calculus of variations arisen from the parameterization are fully explored and revealed in this paper. The conclusion is that Lagrange greatly gen-eralizes the study of the following subjects in the process of the parameterization: the variational problem,the variational equation,the transversality condition,and the principle of least action of mechanics etc. Lagrange's parameterization of his method not only broadens the scope of the study of the calculus of variations,but also gives its significant value in mechanical applications.