David Barton

Applications of algebraic manipulation programs in physics

This paper describes the application to three areas of physics of computer programs that carry out formal algebraic manipulation. The application areas discussed are celestial mechanics, general relativity and quantum electrodynamics. The paper describes typical problems from each of these disciplines which can be solved using algebraic manipulative systems and presents sample programs for the solution of these problems using several algebra systems. For each discipline a review of published work acknowledging the use of algebra programs is presented and the most advanced applications are discussed in detail. In particular the Lie transform, Petrov classification and Kahanes simplification procedure are reviewed from the standpoint of algebra programs. A number of simple examples are used to introduce the reader to the capabilities of an algebra program and a brief review of the technical problems of algebraic manipulation is given. Further applications of such systems to mathematics, chemistry and engineering are briefly mentioned in the text and relevant work is referenced in the bibliography but the main emphasis is placed on applications in theoretical physics. However, the simple examples indicate, and the applications in the physical sciences confirm, that algebra systems are capable of exploitation over a much wider area than is covered in the present review. This review was completed in December 1971.

An algebra system

This paper describes a computing system that enables problems of manipulative algebra involving a number of elementary functions to be simply and efficiently programmed. The system has been designed with particular reference to the problems involved in the explicit calculation of the Riemann tensor and associated quantities.

A review of algebraic manipulative programs and their application

This paper describes the applications area of computer programs that carry out formal algebraic manipulation. The first part of the paper is tutorial and severed typical problems are introduced which can be solved using algebraic manipulative systems. Sample programs for the solution of these problems using several algebra systems are then presented. Next, two more difficult examples are used to introduce the reader to the true capabilities of an algebra program and these are proposed as a means of comparison between rival algebra systems. A brief review of the technical problems of algebraic manipulation is given in the final section

General relativity and the application of algebraic manipulative systems

The paper describes some applications of symbolic algebra systems to problems of general relativity including the derivation of the field equations, the Petrov classification of a metric, and the solution of the field equations in the presence of matter in a simple case. Attention is drawn to the strictly algebraic difficulties encountered in this work.

Problem #4: the Lie transform

The problem proposed here is that of performing a Lie transformation. The Lie transform as described by Deprit (1969) is a means of solving certain sets of ordinary differential equations by repeated approximation and he has applied the technique to the problems of predicting the motion of the Moon and also of artificial satellites. Deprits work contains a remarkable application of symbolic techniques to real problems and it is for this reason that the Lie transform is an interesting calculation.

Algebraic manipulation in Cambridge

Algebraic Manipulation is by no means new to Cambridge. As long ago as 1949, Boys was using an algebra system, specially written by him, for solving problems in theoretical chemistry. The projects that we wish to describe here are not descended directly from this, except that we too are largely motivated by the desire to solve real problems. Before describing what we are doing, a few details of the sort of establishment we have is necessary.