``Stroboscopic'' average

The ``stroboscopic'' average is a technique first proposed by Heinemann and Hoffstätter for the computation of the invariant spin axis $\vec n$. I put the word stroboscopic in quotation mark because the expression is highly unfortunate for a person dealing primarily with a map based theory. In a map based theory, a stroboscopic average would involved recording data not at every turn but once in a while. We assume that Heinemann and Hoffstätter were thinking about equations of motion when they coined the term; their paper confirms the prejudice. A true stroboscopic average is rarely discussed in accelerator theory; for example dispersion can be defined rigorously in the presence of 3 tunes as a stroboscopic skipping over $p$ turns if the longitudinal tune $\nu_{3}$ is nearly equal to $q/p$ where $q$ and $p$ are integers. We will retain the term stroboscopic average by respect for the originators and also to avoid undue confusion.

The assumption of a normal form, whether we can compute it analytically or not, gives us a way to average quantities. Namely the idea is that an average over turns is actually an average over the invariant tori.

Let us start with the expression for the one-turn map assuming a normal form. It is given by Eq. (11.16). Dropping the subscript $f$, we get for the $j$-turn map:

$${T}^{j} = \left({I,A}\right)\circ \left({{m}^{j},{e}^{j\theta \circ {a}^{-1}{L}_{y}}}\right)\circ \left({I,{A}^{-1}}\right)$$

$$= \left({{m}^{j},A\circ {m}^{j}\ {e}^{j\widetilde{\theta }{L}_{y}}\... ...m e}~\widetilde{\theta }= \theta (\vec{J})\circ {a}^{-1}\equiv \theta (\vec{I})$$ (11.21)

If we apply this spin-orbital map to an arbitrary spin ${\vec{s}}_{0}$ at location ${\vec{x}}_{0}$ in orbital space, we get for ${\vec{s}}_{j}$ at turn $j$:

$${\vec{s}}_{j} = {A}_{{\vec{x}}_{j}}\ {e}^{j\widetilde{\theta }{L}_{y}}\ {A}_{{\vec{x}}_{0}}^{-1}\ {\vec{s}}_{0}$$ (11.22)

However in Eq. (11.22), we can see that the role of the initial position is special unlike the case of the normalized space. Indeed in actual space there is no guaranty that the spin axis at some position ${{\vec{x}}_{j}}$ bares any resemblance to the spin axis at the initial point ${{\vec{x}}_{0}}$. Therefore, we invert Eq. (11.22):

$${\vec{s}}_{0} = \ {A}_{{\vec{x}}_{0}}{e}^{-j\widetilde{\theta }{L}_{y}}\ {A}_{{\vec{x}}_{j}}^{-1}\ {\vec{s}}_{j}$$ (11.23)

We now sum and take the limit:

$$\prod = \lim\limits_{j\rightarrow \infty }^{} {1 \over N}\sum\limits_{j= ... ...{A}_{{\vec{x}}_{0}}{e}^{-j\widetilde{\theta }{L}_{y}}\ {A}_{{\vec{x}}_{j}}^{-1}$$ (11.24)

Eq. (11.24) is a little tricky. The problem resides in the index $j$ appearing in the matrix ${A}_{{\vec{x}}_{j}}^{-1}$. To perform the average we use the assumption of an orbital normal form and express the matrix ${A}_{{\vec{x}}_{j}}^{-1}$ in terms of the initial action angle variables $\left({\vec{\psi },\vec{I}}\right)$:

$${A}_{{\vec{x}}_{j}}^{-1} = \sum\limits_{\vec{m}}^{} {\Gamma }_{\vec{m}}^{0}(\vec{I})\ \exp\left({i\vec{m}\cdot \left\{{\vec{\psi }+j\vec{\mu }}\right\}}\right)$$

$$= \sum\limits_{\vec{m}}^{} {\Gamma }_{\vec{m}}(\vec{\psi },\vec{I})\ \exp\left({ij\vec{m}\cdot \vec{\mu }}\right)$$ (11.25)

The matrices ${\Gamma }_{\vec{m}}$ in Eq. (11.25) can be rewritten as three columns vectors:

$${\Gamma }_{\vec{m}} = \left({{\vec{\gamma }}_{1}^{\vec{m}},{\vec{\gamma }}_{2}^{\vec{m}},{\vec{\gamma }}_{3}^{\vec{m}}}\right)$$ (11.26)

The matrix $\Pi$ becomes

$$\Pi = {A}_{{\vec{x}}_{0}}\sum\limits_{\vec{m}}^{} \lim\limits_{j\righta... ...vec{\gamma }}_{3}^{\vec{m}}}\right)\exp\left({ij\vec{m}\cdot \vec{\mu }}\right)$$ (11.27)

In Eq. (11.27), each vector ${\vec{\gamma }}_{k}^{\vec{m}}$ is made of 3 components which transform differently under the effect of ${e}^{-j\widetilde{\theta }{L}_{y}}$. First the second component ${\vec{\gamma }}_{k;2}^{\vec{m}}$ is left invariant by the rotation. The first and third component can be expressed using spin phasors:

$${\vec{\sigma }}_{\pm } = \left({\matrix{\pm i\cr 0\cr 1\cr}}\right)\ \ {\rm w}{\rm h}{\rm ... ...}}{\vec{\sigma }}_{\pm }= {e}^{\mp ij\widetilde{\theta }}{\vec{\sigma }}_{\pm }$$ (11.28)

Therefore we have :

$$\vec{\gamma }= {{\gamma }_{3}-i{\gamma }_{1} \over 2}{\vec{\sigma... ...\over 2}{\vec{\sigma }}_{-}+{\gamma }_{2}\left({\matrix{0\cr 1\cr 0\cr}}\right)$$ (11.29)

The vector in Eq. (11.29) represents any of the three column vectors of Eq. (11.27).

$${e}^{-j\widetilde{\theta }{L}_{y}}{e}^{ij\vec{m}\cdot \vec{\mu }}... ...{-ij\vec{m}\cdot \vec{\mu }}{\gamma }_{2}\left({\matrix{0\cr 1\cr 0\cr}}\right)$$ (11.30)

From Eq. (11.30), we see that the infinite series of Eq. (11.27) will converge zero for all terms such that $\vec{m} \neq 0$ provided we are not sitting on some spin-orbital resonance.

$$\Pi =$$

$$=$$ (11.31)

Remark: The convergence of the stroboscopic average towards a vector $\vec{n}$ satisfying Eq. (11.19) requires that the first two terms in Eq. (11.30) average to zero. This will happen even if the map is on an orbital resonance. Thus the vector must exist even on an isolated orbital resonance. This is actually confirmed perturbatively by the perturbative approach to normal form. It is not clear however if a proper phase advance can be defined since it involves removing resonant terms from the rotation around the normal axis $e_2$-- a process not necessary for the simple existence of $\vec{n}$.