Mathematical Sciences: Symbolic Manipulation in the Calculusof Variations

该项目将开发和实施一个图书馆， SMP语言中的符号操作程序来解决 微积分中出现的代数和计算问题 变化. 所有这些问题最近都被证明是可以解决的 在一个明确计算方法中，实际上， 通过极其繁琐的手工计算。 第一个这样的问题是确定给定的 微分方程组可以从一个 变分原理这个问题有多种形式， 包括直接的问题， 是某个变分问题的欧拉.拉格朗日方程，或者 更困难的问题是找到一个积分因子 在某种意义上，这将使系统等效于一些 欧拉。拉格朗日方程。 再加上这个问题， 拉格朗日量本身的确定，可能在一些 一个特别好的形式，它只依赖于低阶 衍生物. 最后，为计算而开发的算法 微分方程的耗散分解将是 扩展到处理涉及多个独立的方程 变量 建议的操纵程序库将 大大加快了这条研究路线， 数学和数学物理的微分方程。

This project will develop and implement a library of symbolic manipulation programs in the SMP language to solve algebraic and computational problems arising in the calculus of variations. All of these problems recently proven to be solvable in an explicit computational way are, in fact, approachable only through extremely tedious hand calculation. The first such problem is determining whether a given system of differential equations can be derived from a variational principle. This problem comes in various forms, including the direct problem of whether the system as it stands is the Euler.Lagrange equations of some variational problem, or the more difficult question of finding an integrating factor which would make the system equivalent, in some sense, to some Euler.Lagrange equations. Coupled with this question is the determination of the Lagrangian itself, possibly in some particularly nice form that depends only on the lower order derivatives. Finally, the algorithms developed for computing dissipative decomposition of differential equations will be extended to deal with equations involving several independent variables. The proposed library of manipulating programs would significantly speed up this line of research into the systems of differential equations of mathematics and mathematical physics.