ABSTRACT
This chapter suggests that the normal form of the oscillator provides the appropriate alternative. This work is an expository account of the use of normal forms in connection with the study of nonlinear harmonic oscillators. It is written at an elementary level, so as to make the ideas accessible to as large an audience as possible, and is self-contained except for some standard definitions and results from the theory of ordinary differential equations, the variational equation and the existence theorem for flows. The chapter deals with the possible exception of the geometrical bias which is everywhere evident. In fact most of the results can be found scattered throughout the literature; they are simply collected in one place. To convert a harmonic oscillator to normal form requires a series of transformations which, although rewarding in result, is tedious in execution. In fact many important topics of research, such as the reduced phase space of an oscillator with symmetries.
