Geometric aspects of reduction of order

Abstract

Using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry. In particular, we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds. In so doing, some new results which generalise the class of Lie group actions which can be used to solve ordinary differential equations are developed.