A GEOMETRIC TREATMENT OF REDUCTION OF ORDER OF ORDINARY DIFFERENCE-EQUATIONS

We generalize the theory of Lie symmetries of ordinary difference equations to the non-autonomous case. A coordinate-invariant treatment in which solutions are sections of a fibre-bundle is employed. It is shown that Lie symmetries of difference equations must be projectable, which is not the case for differential equations. In fact this result extends to partial difference equations. We also show that a time-like symmetry can be used to reduce a nonautonomous difference equation to the autonomous case. Examples are given of this process and of the reduction of order of a non-autonomous system.