Lie Algebraic Methods

Introduction

In this chapter we shall consider systems of the form

$$ \dot{x} = A(x)x ~(10.1) $$

where $A:\mathbb{R}^{n} \rightarrow \mathfrak{g}$ and $\mathfrak{g}$ is the Lie algebra of a Lie group G. The classical structure theory of Lie groups and Lie algebras (see Appendix B and [1,2]) will be used to decompose the system (10.1) into simpler subsystems in a way which generalises the classical Jordan decomposition of single matrices.