The Classical Adiabatic Theorems in Many Dimensions

Abstract

In the preceding chapter we gave the most important known results on adiabatic invariance for systems of one degree of freedom, in which case there is no need to distinguish between ergodic and integrable systems, and to lowest order (Section 8.1) we needed only the simple one phase averaging theorem of Section 3.1. On the other hand, the adiabatic theorems considered in this chapter have proofs very similar to those of the multiphase averaging results of Chapters 5 and 6. Despite their obvious physical significance, these theorems are rarely correctly stated, and even more rarely correctly proved. We will distinguish the only two cases for which precise theorems exist, though it would be interesting to consider cases intermediate between “integrable” and “ergodic”: