Guide to Non-linear Dynamics in Accelerator Physics/Linear Motion

This chapter provides tools to describe linear motions

Linear dynamics of a single particleEdit

For the case of linear dynamics, the motion can be represented by a 2N x 2N matrix. This matrix maps phase space points into phase space points. Let us represent the initial phase space point by  . The transformation can then be represented by

 .

The matrix   will be symplectic. This means that

 

where

 .

Now, in quantum mechanics, we typically deal with Hermitian operators. These can be diagonalized by orthogonal matrices. With symplectic matrices, we can diagonalize the matrix, but here the transformation matrix will be symplectic. To do this, we find the eigenvectors of M. Let us label these as   The positive and negative eigenmodes are related to each other by

 

We can define the normalization by defining an upper indexed vector

 

Then we find the normalization condition

 

The matrix of eigenvectors

 

is symplectic. The invariants are given in terms of the eigenvectors as

 

Linear Motion in terms of Lie operatorsEdit

We may also describe the one turn map as an operator on x and p. Let us consider the rotation matrix   We may represent this in terms of functions by the Lie operator   This operator acts on the functions x and p in the following way   and   The eigenfunctions of R are given by   with  .   are sometimes referred to as the resonance basis. In the non-linear problems, we will need to compute various operators built out the linear operator. Expanding in terms of the resonance basis will allow us to do these calculations.

2-D phase spaceEdit

Here the one turn map is a 2 x 2 matrix with determinant 1. We can parametrize it by

 

where