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We consider a longitudinal solenoid field with an axial symmetry:
B_z(s)= \left\{\begin{array}{lc} 0, & s<-f/2\\ B_0(s/f+1/2), & -f/2\le s \le f/2\\ B_0, & f/2<s \end{array}\right.\ , }}
where 
is the length of the fringe. The associated vector potentials for 
are
where the terms with 
-functions are necessary in order to satisfy the Maxwell equations, while keeping the axial symmetry. The magnetic field derived from the -function terms in 
and 
are
Thus the changes in momenta due to these fields at 
are
where we have used 
and 
. The equations of motion 
and 
are derived from a Hamiltonian:
D=\frac{B_0}{16fp}(x^2+y^2)(xp_y-yp_x)\left[\delta(s+f/2)-\delta(s-f/2)\right] . }}
The fringe field has at least two effects, linear and nonlinear. The linear effect is caused by the first terms in Eqs. and that are linear in 
and 
. We can expect that such linear effects can be expressed by a model with hard edges sliced along 
, if the number of slices is sufficiently large. Thus here we concentrate on the nonlinear effects that are caused by the -function terms in Eqs. and , or their Hamitonian.
Let us obtain the transformation associated with the nonlinear terms up to the first order of 
. It is expressed as
\exp(:-f/2:)\exp(:-\delta:)\exp(:f:)\exp(:\delta:)\exp(:-f/2:)\ , }}
where 
is a drift-back by a distance 
, and 
is the nonlinear term at , etc. Then the transformation (trans) is approximated as
\begin{pmatrix} x_1\\p_{x1}\\y_1\\p_{y1}\end{pmatrix}= \begin{pmatrix} x_0+b\left(2p_{x0}x_0y_0-p_{y0}(x_0^2-y_0^2)\right)/8p^2\\ p_{x0}+b\left(2p_{x0}p_{y0}x_0-(p_{x0}^2-p_{y0}^2)y_0\right)/8p^2\\ y_0-b\left(2p_{y0}x_0y_0+p_{x0}(x_0^2-y_0^2)\right)/8p^2\\ p_{y0}-b\left(2p_{x0}p_{y0}y_0+(p_{x0}^2-p_{y0}^2)x_0\right)/8p^2 \end{pmatrix}\ , }}
where 
, up to the first order of . The transformation (apptrans) is expressed as 
with a Hamiltonian:
H=-\frac{b}{8p^2}(xp_y-yp_x)(xp_x+yp_y)\ .
}}
It is interesting that the transformation (apptrans) and thus the Hamiltonian (hami) are independent on the length of fringe, .
To solve Hamiltonian eq., it is convenient to use another set of variables:
\begin{pmatrix} r\\ p_r\\ \varphi\\ p_\varphi\end{pmatrix}=\begin{pmatrix} \log(x^2+y^2)/2\\xp_x+yp_y \\ \tan^{-1}(y/x)\\ xp_y-yp_x\end{pmatrix}\ , }} which is generated by a generating function:
G(x,p_r,y,p_\varphi)=\frac{\log(x^2+y^2)}{2}p_r+\tan^{-1}\left(\frac{y}{x}\right)p_\varphi\ . }} Then the Hamiltonian is rewritten as
H=-\frac{b}{8p^2}p_\varphi p_r\ . }}
The transformation with this hamiltonian is simply written as:
\begin{pmatrix}
r_1\\ \varphi_1 \end{pmatrix}=\begin{pmatrix}r_0-bp_\varphi/8p^2\\ \varphi_0-bp_r/8p^2\end{pmatrix}\ ,
}}
where 
and 
are constant.
