Time evolution of $v_{\parallel}$

The time evolution of $v_{\parallel}$ can be obtained by dotting Eq. (13) by $\ensuremath{\boldsymbol{b}}$, which gives

$$\frac{Z e}{c} \frac{\partial \ensuremath{\boldsymbol{A}}}{\partia... ...symbol{b}} \cdot \nabla B - Z e \ensuremath{\boldsymbol{b}} \cdot \nabla \phi .$$ (22)

Noting that

$$\ensuremath{\boldsymbol{b}} \cdot \frac{\partial \ensuremath{\boldsymbol{b}}}{\partial t} = \frac{1}{2} \frac{\partial b^2}{\partial t} = 0,$$ (23)

Eq. (22) is written as

$$\frac{Z e}{c} \frac{\partial \ensuremath{\boldsymbol{A}}}{\partia... ...symbol{b}} \cdot \nabla B - Z e \ensuremath{\boldsymbol{b}} \cdot \nabla \phi .$$ (24)

Using

$$\ensuremath{\boldsymbol{E}} = - \nabla \phi - \frac{1}{c} \frac{\partial \ensuremath{\boldsymbol{A}}}{\partial t},$$ (25)

Eq. (24) is written as

$$m \dot{v}_{\parallel} = - \dot{\ensuremath{\boldsymbol{X}}} \cdot... ...ot \nabla B + Z e \ensuremath{\boldsymbol{b}} \cdot \ensuremath{\boldsymbol{E}}$$ (26)

Noting that the magnetic curture is given by $\ensuremath{\boldsymbol{\kappa}} = - \ensuremath{\boldsymbol{b}} \times \nabla \times \ensuremath{\boldsymbol{b}}$, the above equation is written as

$$m \dot{v}_{\parallel} = m v_{\parallel} \dot{\ensuremath{\boldsym... ...t \nabla B + Z e \ensuremath{\boldsymbol{b}} \cdot \ensuremath{\boldsymbol{E}},$$ (27)

which governing the time evolution of $v_{\parallel}$.