49 - Contact Transformation

Abstract

This chapter describes the contact transformation applicable to first-order and (occasionally) second-order ordinary differential equations. It yields a reformulation which may lead to an exact solution (sometimes in parametric form). By changing variables a different, and sometimes easier, differential equation may be found. Given a relation between three variables ф(x,y,p) = 0 it will be a first-order ordinary differential equation if dy − pdx = 0. If the variables in ф(x,y,p,) = 0 are changed by x = x(X,Y,P), y = y(X,Y,P), and p = p(X,y,P), then the transformed equation ф(X,Y,P) = 0 will also be an ordinary differential equation if dY — PdX = 0. If this is true, then x = x(X,Y,P), y = y(X,Y,P), and p = p(X,y,P) is a contact transformation. Composing two contact transformations or taking the inverse of a contact transformation, results in another contact transformation. Because the identity transformation is also a contact transformation, the set of all contact transformations forms an infinite dimensional topological group.