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Appendix: Details of the normalization pro-
cess for a magnetic bottle

In this appendix we discuss some details of the transformation of a Hamiltonian with a quadratic part (34) into generalized normal form. More specifically, we show how to simplify the Lie operator

\begin{displaymath}
{\cal A}_m(\cdot) %% = \left\{ \cdot,H_2(\VEC{z}) \right\} ...
...\partial}{\partial z_{n+\nu}}(\cdot)
\right)
\right]_{\L _m}
\end{displaymath}

which is adjoint to this particular $H_2$.

The unitary matrix that by a similarity transformation puts the Hamiltonian matrix

\begin{displaymath}
L = J \mbox{Hess}(H_2)
= \left(
\begin{array}{cc}
0_n&\f...
...ts & & 0 & -\omega_n
\end{array} $}& 0_n
\end{array} \right)
\end{displaymath}

into the Jordan normal form $\tilde{L}=MLM^*$ of (35) is

\begin{displaymath}
M = \left( \begin{array}{c}
{\mbox{\protect\boldmath$e$}}_...
...otect\boldmath$e$}}_{2n}^T \right)
\end{array} \right) \quad,
\end{displaymath}

with ${\mbox{\protect\boldmath$e$}}_i$ being the canonical base vectors of ${\bf R}^{2n}$. This formula can easily be derived by performing a certain permutation of the rows and columns first, followed by a transformation similar to (15).

In the new coordinates $\tilde{{\mbox{\protect\boldmath$z$}}}=M{\mbox{\protect\boldmath$z$}}$ the Lie operator takes on the form

\begin{displaymath}
\tilde{{\cal A}}_m
= \left[
\sum_{\nu=1}^l \tilde{z}_{2\n...
...rtial}{\partial\tilde{z}_{n+\nu}}
\right) \right]_{\L _m} \;.
\end{displaymath}

This representation of the Lie operator is advantageous, because here we have collected as many non-zero entries (of the matrix representation) of ${\cal A}_m$ on the diagonal as possible. Only the first sum yields an off-diagonal contribution.

We have not yet made any assumptions about the ordering of the monomials $\tilde{{\mbox{\protect\boldmath$z$}}}^{\mbox{\protect\footnotesize\protect\boldmath$j$}}$ in the basis of $\L _m$. If one chooses the lexicographical ordering [2] of the basis monomials, then the matrix representation of $\tilde{{\cal A}}_m$ becomes an upper diagonal matrix for all $m$, and all the manipulations of ${\cal A}_m$ that are necessary in the course of the normalization procedure (solving linear equations, inverting ${\cal A}_m$, ...) become easier and consume much less computing time.

In the case $l=1$ it is possible to achieve even further simplification by an appropriate ordering of the basis monomials of $\L _m$. One can introduce the so-called magnetic bottle ordering of monomials which results in $\tilde{{\cal A}}_m$ being bidiagonal.


next up previous contents
Next: Bibliography Up: art_ver2 Previous: Concluding remarks   Contents
Martin_Engel 2000-05-25