LIE-BACKLUND AND NOETHER SYMMETRIES WITH APPLICATIONS

New identities relating the Euler-Lagrange, Lie-Backlund and Noether o perators are obtained. Some important results are shown to be conseque nces of these fundamental identities. Furthermore, we generalise an in teresting example presented by Noether in her celebrated paper and pro ve that any Noether symmetry is equivalent to a strict Noether symmetr y, i.e. a Noether symmetry with zero divergence. We then use the symme try based results deduced from the new identities to construct Lagrang ians for partial differential equations. In particular, we show how th e knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lag rangian for the equation. Several examples are given.