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2.2 A particular case: Hill's equation - Betatron functions

The resolution of the homogeneous equation will be performed in a general way in the next section, but only to focus and the the general solution of the perturbated equation. In beam physics a particular case of the homogeneous equation of motion was studied with attention, and a specific formalism has been introduced: the betatron functions. We will introduce it quikly in this section as they will be mentionned later in this document. Removing the dispersive term the trajectory of a particle satisfy the following equation of motion:

$\displaystyle u''+K(s) u =0$

(2.3)

Assuming

, this equation is called Hill's differential equation. If

were a constant (for instance positive), we know that the harmonic solution would be of the form $A \cos(\omega s) +B \sin(\omega s)$ , which can also be written $\tilde{A}\cos\left(\omega s + \tilde{B}\right)$ . This justify the use of an ansatz solution for equation 2.3 of the form:

$\displaystyle u(s) = \sqrt{\epsilon} \sqrt{\beta(s)} \cos (\psi(s) + \psi_0)$

(2.4)

Introducing this form in equation 2.3 will introduce several conditions for equation 2.4 to be a solution. Nevertheless we will not perform this operations here, and refer the reader to the following reference[43]. $\psi$ is defined as the phase advance, and calculated as:

$\displaystyle \psi(s) = \int_0^s \frac{dt}{\beta(t)}$

(2.5)

and two other parameters are introduced in this formalism:

$\displaystyle \alpha(s)$	$\displaystyle =$	$\displaystyle -\frac{1}{2}\frac{d\beta(s)}{ds}$	(2.6)
$\displaystyle \gamma(s)$	$\displaystyle =$	$\displaystyle \frac{1+\alpha(s)^2}{\beta(s)}$	(2.7)

After some algebra[43], it can be shown that the previous sets of equations leads to the following, in which we recognize the equation of an ellipse:

$\displaystyle \gamma u^2 + 2\alpha u u' +\beta u' = \epsilon$

(2.8)

It should not be forgotten here that, all symbols are functions of

, except $\epsilon$ which is an integration constant. Thus, equation 2.8 is the constant of motion to the homogeneous equation, called the Courant-Snyder invariant. As we will further see the the Courant-Snyder invariant $\epsilon$ , is the emittance. $\alpha$ , $\beta$ and $\gamma$ are called Twiss-parameters, defining the shape and orientation of the ellipse. Physical interpretations of these coefficients and their relations with the beam envelope in phase space will be described in the next section. Unfortunatly, we will not go further in the study of betatron function in beam dynamics.

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