Mathematical Sciences: Symbolic Manipulation in the Calculusof Variations

该项目将开发和实现一个SMP语言的符号操作程序库，以解决变分学中出现的代数和计算问题。所有这些最近被证明可以通过显式计算方式解决的问题，实际上只能通过极其繁琐的手工计算来接近。第一个问题是确定一个给定的微分方程组是否可以由变分原理导出。这个问题以各种形式出现，包括系统是否为欧拉的直接问题。一些变分问题的拉格朗日方程，或者更困难的问题找到一个积分因子使系统在某种意义上，与欧拉方程等效。拉格朗日方程。与这个问题相结合的是拉格朗日函数本身的确定，可能以某种特别好的形式只依赖于低阶导数。最后，为计算微分方程的耗散分解而开发的算法将扩展到处理涉及多个自变量的方程。所提出的操作程序库将大大加快对数学和数学物理微分方程系统的研究。

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.