The Extremal Principle

Abstract

A successful research mathematician has mastered a dozen general heuristic principles of large scope and simplicity, which he/she applies over and over again. These principles are not tied to any subject but are applicable in all branches of mathematics. He usually does not reflect about them but knows them subconsciously. One of these principles, the invariance principle was discussed in Chapter I.It is applicable whenever a transformation is given or can be introduced. If you have a transformation, look for an invariant! In this chapter we discuss the extremal principle, which has truly universal applicability, but is not so easy to recognize, and therefore must be trained. It is also called the variational method, and soon we will see why. It often leads to extremely short proofs.