Algebraic manipulations in celestial mechanics


Andre Deprit, Senior Fellow
National Institute of Standards and Technology
Gaithersburg, MD 20899-0001

Ph: (301) 975--2709
Fx: (301) 963--9137

In the world of naive mathematics, we start with the view that a formula is
a text to be transformed.Symbols are strings of characters,
mathematical expressions are concatenations of symbols. Recognizing a
pattern in a phrase,  applying to it a set of rules to make it fit another
pattern, we call that print transposition a calculation.  Going from the
text "(1 - x) (1 + x)" to the text "1 - x^2" is an expansion; the reverse
is a factoring. Mathematics deals with patterns which it calls structures;
computers deal with texts. Symbolic Algebraic Processing  (SAP, for short) 
operates at the intersection between  mathematics and  text processing.. 

        How does a SAP translate a mathematical sentence like "Let x be a
vector"? By installing the symbol x  as a "programming object", i.e., by
attaching to it a type and various operations that make, according to its
mathematical definition, an element in a category of modules over a ring. 
To the hierarchy of mathematical structures 
should correspond lines of ascendancy among programming objects.

        The paper will explain briefly how to revamp a processor of
mathematical texts like Mathematica as a tool of "object oriented
programming"  It will give several examples borrowed from recent
publications. It will emphasize how easy it has become to execute extensive
symbolic calculations through SAP on a portable computer, the equivalent
today of the back of an envelope.