A more compact form of the time evolution of $v_{\parallel}$

The time evolution of $v_{\parallel}$ is given by Eq. (27), i.e.,

$$m v_{\parallel} \dot{v}_{\parallel} = m v_{\parallel}^2 \dot{\ens... ..._{\parallel} Z e \ensuremath{\boldsymbol{b}} \cdot \ensuremath{\boldsymbol{E}},$$ (74)

which involves the term $\dot{\ensuremath{\boldsymbol{X}}} \cdot \ensuremath{\boldsymbol{\kappa}}$. Next we try to simplify this term. Using Eq. (71), the term is written as

$$m v_{\parallel}^2 \dot{\ensuremath{\boldsymbol{X}}} \cdot \ensuremath{\boldsymbol{\kappa}} = - \frac{m v_{\parallel}^2 Z e}{m \Omega B^{\star}} \left( \ensure... ...ldsymbol{B}} \times \mu \nabla B \right) \cdot \ensuremath{\boldsymbol{\kappa}}$$

$$= \frac{Z e v_{\parallel}^2}{\Omega B^{\star}} \left( \ensuremath{\... ...( \ensuremath{\boldsymbol{b}} \times \nabla \times \ensuremath{\boldsymbol{b}})$$

$$= \frac{Z e v_{\parallel}^2}{\Omega B^{\star}} \left( B \ensuremath... ...suremath{\boldsymbol{b}} \mu \nabla B \cdot \ensuremath{\boldsymbol{b}} \right)$$

Using this in Eq. (74) gives

$$m v_{\parallel} \dot{v}_{\parallel} = \frac{Z e v_{\parallel}^2}{... ...\parallel} Z e \ensuremath{\boldsymbol{b}} \cdot \ensuremath{\boldsymbol{E}},$$

which can be arranged as

$$m v_{\parallel} \dot{v}_{\parallel} = \left[ \frac{v_{\parallel}^... ...l{b}} + v_{\parallel} \ensuremath{\boldsymbol{b}} \right] \cdot \mu \nabla B,$$

which, after some straightforward algebras, can be arranged into the following forms

$$m v_{\parallel} \dot{v}_{\parallel} = \ensuremath{\boldsymbol{v}}... ...l}^{\star} \cdot \left( Z e \ensuremath{\boldsymbol{E}} - \mu \nabla B \right),$$ (75)

which agrees with Eq. (15) in Todo's paper[4].