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

This chapter provides tools to describe linear motions

Linear dynamics of a single particle
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 $$\vec z_0$$. The transformation can then be represented by

$$\vec z = M \vec z_0$$.

The matrix $$M$$ will be symplectic. This means that

$$M^TJM=J$$

where

$$J=\left(\begin{matrix}0&1&0&0&0&0\\ -1&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&-1&0&0&0\\ 0&0&0&0&0&1\\ 0&0&0&0&-1&0\end{matrix}\right)$$.

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 $$v_{\pm 1,\pm 2,\pm 3}$$  The positive and negative eigenmodes are related to each other by

$$v_{-k} = iv_k^*$$

We can define the normalization by defining an upper indexed vector

$$v^j=-i{\rm sgn}(j)v_j^*$$

Then we find the normalization condition

$$v^jv_k=\delta_{jk}$$

The matrix of eigenvectors

$$U=(v_1\ v_{-1} v_2\ v_{-2}\ v_3\ v_{-3})$$

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

$$G_a = -J(v_a v_a^\dagger+v_a^*v_a^T)J$$

Linear Motion in terms of Lie operators
We may also describe the one turn map as an operator on x and p. Let us consider the rotation matrix $$\begin{pmatrix}x\\ p\end{pmatrix} = \begin{pmatrix}cos\mu & \sin\mu\\ -\sin\mu & \cos\mu\end{pmatrix}\begin{pmatrix}x_0\\ p_0\end{pmatrix}$$ We may represent this in terms of functions by the Lie operator $$R=e^{\frac{\mu}{2}:x^2+p^2:}$$ This operator acts on the functions x and p in the following way $$Rx=\cos\mu x-\sin\mu p$$ and $$Rp=\sin\mu p+\cos\mu x$$ The eigenfunctions of R are given by $$h_\pm=x\mp i p$$ with $$R h_\pm = e^{\pm i\mu} h_\pm $$. $$h_\pm$$ 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
Here the one turn map is a 2 x 2 matrix with determinant 1. We can parametrize it by

$$M_x = I \cos\mu + J_x \sin\mu $$

where

$$J_x = \left(\begin{matrix} \alpha & \beta \\ -\gamma & -\alpha\end{matrix}\right)$$