Unperturbed orbit integration----not finished------

For the ease of notation, in the following we drop the zero superscript on the unperturbed orbit. And to distinguish instantaneous and the initial value of orbit, we add a prime to $\ensuremath{\boldsymbol{X}}$ and $v_{\parallel}$ to denote the instantaneous value. Integrating along the unperturbed orbit, Eq. (122) is written as

$$g^{(1)} ( \ensuremath{\boldsymbol{X}}, v_{\parallel}, y) = \int_{- \infty}^0 \mathcal{L}^{(1)} (\tau) d \tau .$$ (134)

with the boundary condition

$$\ensuremath{\boldsymbol{X}}' (\tau = 0) = \ensuremath{\boldsymbol{X}},$$ (135)

$$v_{\parallel}' (\tau = 0) = v_{\parallel},$$ (136)

and the value of the conserved magnetic moment is determined by $\mu = y / B ( \ensuremath{\boldsymbol{X}})$. Using the expression of $\mathcal{L}^{(1)}$ in Eq. (123), Eq. (134) is written as

$$g^{(1)} = \int_{- \infty}^0 \left[ \frac{Z e}{c} \ensuremath{\boldsymbol{A}... ...h{\boldsymbol{X}}}' - \mu_0 B_{\parallel}^{(1)} - Z e \phi^{(1)} \right] d \tau$$

$$=$$

$$B^{(1)} (\psi, \theta, \varphi, t) = \hat{B}^{(1)} (\psi, \theta) \exp \left[ - i \left( \omega t + n \varphi \right) \right]$$ (137)

$$\dot{\psi} = \dot{\ensuremath{\boldsymbol{X}}} \cdot \nabla \psi$$ (138)

$$\dot{\theta} = \dot{\ensuremath{\boldsymbol{X}}} \cdot \nabla \theta$$ (139)

$$\dot{\varphi} = \dot{\ensuremath{\boldsymbol{X}}} \cdot \nabla \varphi$$ (140)

$$\varphi' (t + l t_p) = \varphi' (t) + 2 l \pi$$ (141)