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.
