Generalized Symmetries

Abstract

The previous sections discussed the classical method, the non-classical method, and some extensions of Lie’s theory. We demonstrated the classification and solution of some types of ordinary as well as partial differential equations. Although very general, these methods mainly consider the geometric aspects of the transformations related to the equations. Essentially, we discussed invariance under point transformations and demonstrated that some kind of contact transformations are related to point transformations in case of first-order partial differential equations. However, it turns out that much wider classes of transformations leave differential equations invariant, including those considered by Lie.