In this chapter we can begin our study of the invariant calculus. The concept from which all else derives is that of a moving frame. We use the definition and construction as detailed by Fels and Olver (1998, 1999). Although the term ‘moving frame’, or ‘repère mobile’ is associated with Èlie Cartan (1953), the idea was used, albeit implicitly, long before. A pre-Cartan history of the subject is given by Akivis and Rosenfeld (1993), and the Fels and Olver papers have a more recent historical overview. The definition of a moving frame used here has the major advantage that it can be applied to both smooth and discrete problems. In particular, there is no need for any of the paraphernalia of Differential Geometry such as exterior calculus, frame bundles and connections.

Moving frames

The original problem solved by moving frames was the equivalence problem, ‘when can two surfaces be mapped one to the other, under a coordinate transformation of a particular type?’ It turns out there are many problems which can be formulated this way. One is the classification problem of differential equations. If you have a differential equation to solve and a database of solved equations, it is only sensible to ask, is there a coordinate transformation that takes my equation to one of the solved ones? Viewing differential equations as surfaces in (x, u, ux, uxx, …) space, you might then apply moving frame theory.