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

This chapter provides tools to describe linear motions

## Linear dynamics of a single particle[edit]

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 operators[edit]

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 space[edit]

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

where