Separation of perturbed distribution into adiabatic and non-adiabatic parts

Write $f_1$ as

$$f_1 = P_{\varphi}^{(1)} \frac{\partial F}{\partial P_{\varphi 0}}... ...epsilon_0} - \mu_0 \frac{B_1}{B_0} \frac{\partial F}{\partial \mu_0} + h^{(1)},$$ (113)

and substitute this into Eq. (112), giving an equation of $h^{(1)}$,

$$\frac{D h^{(1)}}{D t} = - \left( \frac{\partial \mathcal{L}}{\par... ...\partial P_{\varphi 0}} + L_t^{(1)} \frac{\partial F}{\partial \varepsilon_0} .$$ (114)

Eq. (114) agrees with Eq. (35) in Porcelli's paper[1]. For notation convenience, we define $\mathcal{L}^{(1)}$ as

$$\mathcal{L}^{(1)} = \left( \frac{Z e}{c} \ensuremath{\boldsymbol{... ...\cdot \dot{\ensuremath{\boldsymbol{X}}}^{(0)} - \mu_0 B^{(1)} - Z e \phi^{(1)},$$ (115)

which can be called ``perturbed Lagrangian'' (I do not care the name of $\mathcal{L}^{(1)}$, and $\mathcal{L^{(1)}}$ is only a notation without any physical meaning since I do not need this meaning to derive anything). Since we are considering toroidal symmetrical case, different toroidal harmonics of perturbation are independent. Thus we can consider a single toroidal harmonic, i.e, the $\varphi$ dependence of components of $\ensuremath{\boldsymbol{A}}^{(1)}$ and $\ensuremath{\boldsymbol{b}}^{(1)}$ is $\exp (- i n \varphi)$. Further due to that the equilibrium is time-independent, we can consider a single time harmonic, i.e., the time dependence of the perturbation is $\exp (- i \omega t)$. Then, using Eq. (115), Eq. (99) is written as

$$\left( \frac{\partial \mathcal{L}}{\partial \varphi} \right)^{(1)} = - i n \mathcal{L}^{(1)},$$ (116)

and $\mathcal{L}_t^{(1)}$ in Eq. (98) is written as

$$\mathcal{L}_t^{(1)} = - i \omega \mathcal{L}^{(1)} .$$ (117)

Using Eqs. (116) and (117), Eq. (114) is written as

$$\frac{D h^{(1)}}{D t} = i n \mathcal{L}^{(1)} \frac{\partial F}{\... ...hi 0}} - i \omega \mathcal{L}^{(1)} \frac{\partial F}{\partial \varepsilon_0} .$$ (118)

Define

$$\omega_{\star} \equiv \frac{\partial F}{\partial P_{\varphi 0}} / \frac{\partial F}{\partial \varepsilon_0},$$ (119)

then Eq. (118) is written as

$$\frac{D h^{(1)}}{D t} = - i \left( \omega - n \omega_{\star} \right) \frac{\partial F}{\partial \varepsilon_0} \mathcal{L}^{(1)} .$$ (120)

Eq. (120) agrees with Eq. (40) in Porcelli's paper. Define $g^{(1)}$ as

$$h^{(1)} = - i \left( \omega - n \omega_{\star} \right) \frac{\partial F}{\partial \varepsilon_0} g^{(1)},$$ (121)

then $g^{(1)}$ satisfies (note that $\omega_{\star}$ and $\partial F / \partial \varepsilon_0$ are both functions of constants of motion, thus can be taken out of the orbit integration)

$$\frac{D g^{(1)}}{D t} = \mathcal{L}^{(1)} .$$ (122)