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2.3 General resolution of the perturbated linear equation of motion

2.3.0.1 Vectorial form

Boundaries conditions must be added for the problem to be solvable:

$$f(s) = u''+ K(s).u$$

$$u(s_0) = u_0$$

$$u'(s_0) = u'_0$$

For the resolution we will use the general method of a nth degree linear differential equation using the Resolvent matrix. This method is very nice and has the advantage of giving a result directly on matrix form. To get started, let's write equation 2.2 in vectorial formalism:

$$\begin{pmatrix}u'\\ u''\\ \end{pmatrix} = \begin{pmatrix}0 & 1\\ ... ...\begin{pmatrix}u\\ u'\\ \end{pmatrix} + \begin{pmatrix}0\\ f(s)\\ \end{pmatrix}$$ (2.9)

Now, we define $X=(u \;u')^{T}$, and equation 2.9 reduces to the following general linear differential equation:

$$X'(s)=A(s)X(s) + B(s)$$ (2.10)

The method we will further describe can apply to any vector $X=(u\; u'\; u''\; ...\; u^{(n)})^T$ in a more general context of a n$^{th}$ degree differential equation.

2.3.0.2 Homogeneous equation

We first solve the Homegeneous Equation:

$$X'(s)=A(s)\!X(s)$$ (2.11)

and define the resolvent matrix $R(s) = [C_1\: ...\: C_n]$ where $C_i=(x_i(s)\; x_i'(s)\; ...\; x_i^{(n)}(s))^T$ and $(x_i)_{i\in\left[1;n\right]}$ is a base of the vectorial space of solutions. These solutions are called principal solutions. As we have $\forall i\ C_i'=A.C_i$ then

$$R(s)'=A(s).R(s)$$

and the general solution will be:

$$X(s)=R(s).\Omega_0$$

where $\Omega_0=(u_0\; u'_0\; ... \; u^{(n)}_0)^T$, is a constant vector defined by the boundary conditions.

2.3.0.3 Particular solution by variation of constant parameter

Let's find a solution that can be written $X(s)=R(s).\Omega(s)$ and solve the general system.

$$X' = AX+B$$ (2.12)

$$X = R.\Omega$$ (2.13)

Replacing 2.13 in the general equation 2.12, and remembering that $R'=AR$ we have:

$$R\Omega'=B$$

The invertibility of R, given by the fact that it contains base vectors leads us to:

$$\Omega(s)=\int^{s}_{s_0} R^{-1}(t)B(t)dt$$

and the particular solution is defined.

2.3.0.4 General Solution

We eventually find the general solution as the sum of the particular solution and the homogeneous solution:

$$X(s) = R(s).(\Omega_0+\Omega(s))$$ (2.14)

2.3.0.5 Going back to two dimensions

In two dimension, the resolvent matrix $R(s) = [C_1\: ...\: C_n]$ , can be written, without loss of generality:

$$R(s)=\begin{pmatrix} C(s) & S(s)\\ C'(s)& S'(s)\\ \end{pmatrix}$$

$$R^{-1}(s)=\begin{pmatrix} S'(s) & -S(s)\\ -C'(s)& C(s)\\ \end{pmatrix}$$

Now, in our particular case, $f(s)=\frac{1}{\rho_0}(s) \:\delta$

And by assuming $\delta$ stay constant between $s_0$ and $s$:

$$\displaystyle\Omega(s)=\int^{s}_{s_0} R^{-1}(t) \begin{pmatrix}0\... ...gin{pmatrix} \frac{-S}{\rho_0}(t) \\ \frac{C}{\rho_0}(t) \\ \end{pmatrix} dt$$

We can now define D, the dipersion function, from:

$$R(s).\Omega(s)=\delta \! \begin{pmatrix} D(s) \\ D'(s) \\ \end{pmatrix}$$

So that the dispersion function is:

$$D(s)=\int^{s}_{s_0} \frac{1}{\rho_0}(t)\left[S(s)C(t)-C(s)S(t) \right]dt$$ (2.15)

Then we can add a new initial parameter, $\delta = \frac{\delta p}{p}$ and eventually the general solution 2.15 can be written:

$$\begin{pmatrix} u \\ u'\\ \delta\\ \end{pmatrix}= \begin{pmatr... ... \end{pmatrix}. \begin{pmatrix} u(s_0) \\ u'(s_0)\\ \delta_0\\ \end{pmatrix}$$

2.3.0.6 Particular case of $K(s)=k$ constant

It is straightforward to find principal solutions for equation 2.9.

For $k$	<	0		$C(s)$	=	$$\cosh \sqrt{-k}\left(s-s_0\right)$$		$S(s)$	=	$$\frac{\sinh \sqrt{-k}\left(s-s_0\right) }{\sqrt{-k} }$$
For $k$	>	0		$C(s)$	=	$$\cos \sqrt{k}\left(s-s_0\right)$$		$S(s)$	=	$$\frac{\sin \sqrt{k}\left(s-s_0\right)}{\sqrt{k}}$$
For $k$	=	0		$C(s)$	=	$1$		$S(s)$	=	$\left(s-s_0\right)$

These results close our section on general beam dynamics to now focus on matrix formalism and beam optics. Indeed, the case $K=k$ is a good approximation for most accelerator components, and the distinction between the cases $k<0$, $k>0$ and $k=0$ respectively reflects the distinction between defocusing elements, focusing elements, and a simple drift.

Emmanuel Branlard
http://emmanuel.branlard.free.fr/