opalt.asciidoc

\frac{\mathrm{d}\gamma\mathbf{v}}{\mathrm{d}t} =   \frac{q}{m}[\mathbf{E}_{ext} + \mathbf{E}_{sc} + \mathbf{v} \times (\mathbf{B}_{ext} + \mathbf{B}_{sc})]
\frac{\mathrm{d}}{\mathrm{d} t} = \frac{\mathrm{d}}{\mathrm{d}\mathbf{s}}\frac{\mathrm{d}\mathbf{s}}{\mathrm{d} t} = \mathbf{\beta}c\frac{\mathrm{d}}{\mathrm{d}\mathbf{s}}.
\mathbf{v} =\left(\begin{array}{c}
    X \\
    Y \\
    Z
  \end{array}\right),
\qquad
\mathcal{W}=\mathcal{S}\mathcal{T}\mathcal{U}
\mathcal{S}=\left(\begin{array}{ccc}
    \cos\Theta &  0 &  \sin\Theta \\
    0         &  1 &   0 \\
    -\sin\Theta &  0 &  \cos\Theta
  \end{array}\right),
\quad
\mathcal{T}=\left(\begin{array}{ccc}
    1 &  0        &  0 \\
    0 &  \cos\Phi &  \sin\Phi \\
    0 & -\sin\Phi &  \cos\Phi
  \end{array}\right),
\quad
\mathcal{U}=\left(\begin{array}{ccc}
    \cos\Psi & -\sin\Psi &  0 \\
    \sin\Psi &  \cos\Psi &  0 \\
    0        &  0        &  1
  \end{array}\right).
\mathbf{v}_i = \mathcal{W}_{i-1}\mathbf{r}_i + \mathbf{v}_{i-1}, \qquad
\mathcal{W}_i = \mathcal{W}_{i-1}\mathcal{S}_i,
\begin{aligned}
        \nabla \cdot \mathbf{B} & = 0 \Rightarrow \exists \quad \mathbf{A} \quad \text{with} \quad \mathbf{B} = \nabla \times \mathbf{A} \\
        \nabla \times \mathbf{B} & = 0 \Rightarrow \exists \quad V \quad \text{with} \quad \mathbf{B} = - \nabla V
    \end{aligned}
\begin{aligned}
        B_x & = - \frac{\partial V}{\partial x} = \frac{\partial A_s}{\partial y} \\ \\
        B_y & = - \frac{\partial V}{\partial y} = - \frac{\partial A_s}{\partial x}
    \end{aligned}
\tilde{A} (z) = \sum_{n=0}^{\infty} \kappa_n z^n, \quad \kappa_n = \lambda_n + i \mu_n
\begin{aligned}
        x & = r \cos \varphi \quad y = r \sin \varphi  \\
        z^n & = r^n ( \cos n \varphi + i \sin n \varphi )
    \end{aligned}
\begin{aligned}
        V(r, \varphi) & = \sum_{n=0}^{\infty} r^n ( \mu_n \cos n \varphi + \lambda_n \sin n \varphi ) \\
        A_s (r, \varphi) & = \sum_{n=0}^{\infty} r^n ( \lambda_n \cos n \varphi - \mu_n \sin n \varphi )
    \end{aligned}
\begin{aligned}
        B_{\varphi} & = - \frac{1}{r} \frac{\partial V}{\partial \varphi} = - \sum_{n=0}^{\infty} n r^{n-1} ( \lambda_n \cos n \varphi - \mu_n \sin n \varphi ) \\
        B_r & = - \frac{\partial V}{\partial r} = - \sum_{n=0}^{\infty} n r^{n-1} ( \mu_n \cos n \varphi + \lambda_n \sin n \varphi )
    \end{aligned}
b_n = - \frac{n \lambda_n}{B_0} r_0^{n-1} \qquad a_n = \frac{n \mu_n}{B_0} r_0^{n-1}
\begin{aligned}
        B_{\varphi}(r, \varphi) & = B_0 \sum_{n=1}^{\infty} ( b_n \cos n \varphi+ a_n \sin n \varphi ) \left( \frac{r}{r_0} \right) ^{n-1} \\
        B_r (r, \varphi) & = B_0 \sum_{n=1}^{\infty} ( -a_n \cos n \varphi+ b_n \sin n \varphi ) \left( \frac{r}{r_0} \right) ^{n-1}
    \end{aligned}
\begin{aligned}
        V & = \sum_{n=1}^{\infty} V_n (r,z) \sin n \varphi \\
        V_n & = \sum_{k=0}^{\infty} C_{n,k}(z) r^{n+2k}
    \end{aligned}
\nabla ^2 V = 0 \Rightarrow \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial V_n}{\partial r} \right) + \frac{\partial^2 V_n}{\partial z^2} = \frac{n^2 V_n}{r^2} = 0
V_n  = \sum_{k=0}^{\infty} C_{n,k}(z) r^{n+2k}
\Rightarrow  \sum_{k=0}^{\infty} \left[ r^{n+2(k-1)} \left[ (n+2k)^2 - n^2 \right] C_{n,k}(z) + r^{n+2k} \frac{\partial^2 C_{n,k}(z)}{\partial z^2} \right] = 0
C_{n,k}(z) = - \frac{1}{4k(n+k)} \frac{d^2 C_{n,k-1}} {dz^2}, k = 1,2, \ldots
C_{n,k} (z) = (-1)^k \frac{n!}{2^{2k} k! (n+k)!} \frac{d^{2k} C_{n,0}(z)}{dz^{2k}}
V_n = - \left( \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!}{2^{2k} k! (n+k)!} C_{n, 0}^{(2k)}(z) r^{2k} \right) r^n
\begin{aligned}
        B_r & = \sum_{n=1}^{\infty} g_{rn} r^{n-1} \sin n \varphi \\
        B_{\varphi} & = \sum_{n=1}^{\infty} g_{\varphi n} r^{n-1} \cos n \varphi
    \end{aligned}
\begin{aligned}
        g_{rn} (r,z) & = \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n! (n+2k)}{2^{2k} k! (n+k)!} C_{n,0}^{(2k)}(z)r^{2k} \\
        g_{ \varphi n} (r,z) & = \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!n}{2^{2k} k! (n+k)!} C_{n,0}^{(2k)}(z)r^{2k}
    \end{aligned}
\begin{aligned}
        B_z & = - \frac{\partial V}{\partial z} =  \sum_{n=1}^{\infty} g_{zn} r^n \sin n \varphi \\
        g_{zn} & = \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!}{2^{2k} k! (n+k)!} C_{n,0}^{2k+1} r^{2k}
    \end{aligned}
\begin{aligned}
        B_{r,n} & = - \frac{\partial V_n}{\partial r} \sin n \varphi = g_{rn} r^{n-1} \sin n \varphi \\
        B_{\varphi,n} & = - \frac{n}{r} V_n \cos n \varphi = g_{\varphi n} r^{n-1} \cos n \varphi \\
        B_{z,n} & = - \frac{\partial V_n}{\partial z} \sin n \varphi = g_{zn} r^{n} \sin n \varphi
    \end{aligned}
\begin{aligned}
        C_{n,0}(z) & = \frac{G_0}{1+exp[P(d(z))]}, \quad G_0 = \frac{B_0}{r_0^{n-1}} \\
        P(d) & = C_0 + C_1 \left( \frac{d}{\lambda} \right) + C_2 \left( \frac{d}{\lambda} \right)^2 + \ldots + C_{k-1} \left( \frac{d}{\lambda} \right)^{k-1}
    \end{aligned}
\begin{aligned}
        \nabla \cdot \mathbf{B} & = \frac{1}{h_s} \left[ \frac{\partial (h_s B_x )}{\partial x} + \frac{\partial B_s}{\partial s} + \frac{\partial (h_s B_z )}{\partial z} \right] \\ \\
        \nabla \times \mathbf{B} & = \frac{1}{h_s} \left[ \frac{\partial B_z}{\partial s} -  \frac{\partial (h_s B_s )}{\partial z} \right] \hat{\mathbf{x}} + \left[ \frac{\partial B_x}{\partial z} - \frac{\partial B_z}{\partial x} \right] \hat{\mathbf{s}} + \frac{1}{h_s} \left[ \frac{\partial (h_s B_s)}{\partial x} - \frac{\partial B_x}{\partial s} \right] \hat{\mathbf{z}}
\end{aligned}
b_z (z) = B_z (-z) \qquad B_x (z) = - B_x (-z) \qquad B_s (z) = - B_s (-z)
\begin{aligned}
         B_z & =   \sum_{i,k=0}^{\infty} b_{i,k} x^i z^{2k} \\
         B_x & = z \sum_{i,k=0}^{\infty} a_{i,k} x^i z^{2k} \\
         B_s & = z \sum_{i,k=0}^{\infty} c_{i,k} x^i z^{2k}
\end{aligned}
\frac{\partial}{\partial x} \left( (1+x/ \rho) B_x \right) + \frac{\partial B_s}{\partial s} + (1+x/ \rho) \frac{\partial B_z}{\partial z} = 0
\begin{aligned}
            \frac{\partial B_z}{\partial s} & = (1+x/ \rho) \frac{\partial B_s}{\partial z}  \\ \\
            \frac{\partial B_x}{\partial z} & = \frac{\partial B_z}{\partial s}   \\ \\
            \frac{\partial B_x}{\partial s} & = \frac{\partial}{\partial x} \left( (1+x/ \rho) B_s \right)
\end{aligned}
\sum_{i,k=0}^{\infty} a_{i,k} (2k+1) x^i z^{2k} = \sum_{i,k=0}^{\infty} b_{i,k} i x^{i-1} z^{2k}
a_{i,k} = \frac{i+1}{2k+1} b_{i+1,k}
\begin{aligned}
        \sum_{i,k=0}^{\infty} \partial_s b_{i,k} x^i z^{2k} & = \left( 1+ \frac{x}{\rho} \right) \sum_{i,k=0}^{\infty} c_{i,k} (2k+1) x^i z^{2k} \\
        \Rightarrow c_{i,k}  & + \frac{1}{\rho} c_{i-1,k} = \frac{1}{2k+1} \partial_s b_{i,k}
    \end{aligned}
\sum_{i,k=0}^{\infty} \left[ \frac{i+1}{\rho} c_{i,k} x^i + c_{i,k} i x^{i-1} \right] z^{k+1} = \sum_{i,k=0}^{\infty} \partial_s a_{i,k} x^i z^{2k}
\Rightarrow \frac{\partial_s a_{i,k}}{i+1} = \frac{1}{\rho} c_{i,k} + c_{i+1,k}
\sum_{i,k=0}^{\infty} \left[ \frac{a_{i,k} x^i z^{2k+1}}{\rho}  + i a_{i,k} x^{i-1} z^{2k+1} + \frac{i a_{i,k} x^i z^{2k+1}}{\rho}  + \partial_s c_{i,k} x^i z^{2k+1} + 2kb_{i,k}x^i z^{2k-1} \right] = 0
\Rightarrow \partial_s c_{i,k} + \frac{2(k+1)}{\rho} b_{i-1,k+1} + 2(k+1) b_{i,k+1} + \frac{1}{\rho} a_{i,k} + (i+1) a_{i+1,k} + \frac{1}{\rho} a_{i,k} = 0
\partial_s c_{i,k} + \frac{(i+1)^2}{\rho (2k+1)} b_{i+1,k} + \frac{(i+1)(i+2)}{2k+1} b_{i+2,k} + \frac{2(k+1)}{\rho} b_{i-1,k+1} + 2(k+1) b_{i,k+1} = 0
B_z(z=0) = B_{0,0} + B_{1,0}x + B_{2,0} x^2 + B_{3,0} x^3 + \ldots
\rho (s) \propto B(z=0, x=0, s) = B_x (z=0,x=0) = b_{0,0}(s)
B_z (x, z=0, s) = T(x) S(s)
\psi = z f_0(x,s) + \frac{z^3}{3!} f_1(x,s) + \frac{z^5}{5!} f_3(x,s) + \ldots
\frac{\partial}{\partial x} \left( h_s \frac{\partial \psi}{\partial x} \right) + \frac{\partial}{\partial s} \left( \frac{1}{h_s} \frac{\partial \psi}{\partial s} \right) + \frac{\partial}{\partial z} \left( h_s \frac{\partial \psi}{\partial z} \right) = 0
\sum_{n=0} \frac{z^{2n+1}}{(2n+1)!} \left[ \partial_x^2 f_n(x,s) + \partial_s^2 f_n(x,s) \right] + \sum_{n=1} f_n(x,s) \frac{z^{n-1}}{(n-1)!} = 0
f_{n+1}(x,s) = - \left( \partial_x^2 + \partial_s^2 \right) f_n(x,s)
\begin{aligned}
        f_n(x,s) & = (-1)^n \left( \partial_x^2 + \partial_s^2 \right)^n f_0(x,s) \\ \\
        f_n(x,s) & = (-1)^n \sum_{i=0}^n \binom{n}{i}T^{(2i)}(x) S^{(2n-2i)}(s)
    \end{aligned}
\left( \frac{1}{\rho h_s} \partial_x + \partial_x^2 + \partial_z^2 + \frac{\partial_s^2}{h_s^2} \right) \psi = 0
\begin{aligned}
        f_{n+1}(x,s) & = - \left[ \frac{1}{\rho + x} \partial_x + \partial_x^2 + \frac{\partial_s^2}{(1+x/ \rho)^2} \right] f_n(x,s) \\ \\
        f_{n+1}(x,s) & = (-1)^n \left[ \frac{1}{\rho + x} \partial_x + \partial_x^2 + \frac{\partial_s^2}{(1+x/ \rho)^2} \right]^n f_0(x,s)
    \end{aligned}
\left[ \frac{1}{\rho h_s} \partial_x + \partial_x^2 + \partial_z^2 + \frac{\partial_s^2}{h_s^2} + \frac{x}{\rho^2 h_s^3} (\partial_s \rho) \partial_s \right] \psi = 0
\begin{aligned}
        f_{n+1}(x,s) & = - \left[ \frac{\partial_x}{\rho h_s} + \partial_x^2 + \partial_z^2 + \frac{1}{h_s^2}\partial_s^2 + \frac{x}{\rho^2 h_s^3} (\partial_s \rho) \partial_s \right] f_n(x,s) \\ \\
        f_{n}(x,s) & = (-1)^n \left[ \frac{\partial_x}{\rho h_s} + \partial_x^2 + \partial_z^2 + \frac{\partial_s^2}{h_s^2} + \frac{x}{\rho^2 h_s^3} (\partial_s \rho) \partial_s \right]^n f_0(x,s)
    \end{aligned}
\rho(s) = B_0 \times S(s)