The Application of Lie Algebra Techniques to Beam Transport Design

The techniques described here can be used for beam line design in quite general circumstances. We are introducing them for a final focus system design in linear col_ . liders because this is where we have applied them, and to provide a specific context for our discussion. Other optical systems may require modifications or extensions to the methods we introduce here. We have kept the formalism and mathematics to a bare minimum, hoping to simplify the presentation and clarify its connection with other methods. The territory we sketch is the tip of the iceberg of Lie algebraic methods. In the last section, we describe briefly a broader context, though the interested reader will need to consult the extensive literature on this subject.r As far as we know, the particulars we present here are original; however, the essence of the method comes from Alex Dragt and collaborators.