(56) | |||

(57) |

The electric force is assumed to be comparable to the magnetic force. To keep track of the order of the various quantities, we introduce the parameter as a book-keeping device, and make the substitution , as well as . The parameter is set to unity in the final answer.

In order to make use of the technique described in the previous section, we write the
dynamical equations in first-order differential form,

and seek a change of variables,

such that the new guiding centre variables and are free of oscillations along the particle trajectory. Here, is a new independent variable describing the phase of the gyrating particle. The functions and represent the gyration radius and velocity, respectively. We require periodicity of these functions with respect to their last argument, with period , and with vanishing mean:

Here, the angular brackets refer to the average over a period in .

The equation of motion is used to determine the coefficients in the expansion
of
and :

(63) | |||

(64) |

The dynamical equation for the gyrophase is likewise expanded, assuming that ,

(65) |

To each order in , the evolution of the guiding centre position
and velocity
are determined by
the solubility conditions for the equations of motion (58)-(59) when
expanded to that order.
The oscillating components of the equations of motion determine the
evolution of the gyrophase. Note that the velocity equation
(58) is *linear*. It follows that, to all orders in , its solubility condition is simply

(66) |

To lowest order [*i.e.*,
], the momentum equation
(59) yields

(68) |

(69) |

where all quantities are evaluated at the guiding-centre position . The perpendicular component of the velocity, , has the same form (39) as for uniform fields. Note that the parallel velocity is undetermined at this order.

The integral of the oscillating component of Eq. (67) yields

(71) |

(72) |

where is the initial phase. Note that the amplitude of the gyration velocity is undetermined at this order.

The lowest order oscillating component of the velocity equation (58) yields

(75) |

The gyrophase average of the first-order [*i.e.*, ]
momentum equation (59) reduces to

(77) |

(78) |

The averages are specified by

(79) |

(80) |

The coefficient of in the above equation,

(81) |

(82) |

(83) |

The first-order guiding centre equation of motion reduces to

Here, use has been made of Eq. (70) and . The component of Eq. (84) perpendicular to the magnetic field determines the first-order perpendicular drift velocity:

Note that the first-order correction to the parallel velocity, the parallel drift velocity, is undetermined to this order. This is not generally a problem, since the first-order parallel drift is a small correction to a type of motion which already exists at zeroth-order, whereas the first-order perpendicular drift is a completely new type of motion. In particular, the first-order perpendicular drift differs fundamentally from the drift, since it is not the same for different species, and, therefore, cannot be eliminated by transforming to a new inertial frame.

We can now understand the motion of a charged particle as it moves through slowly varying electric and magnetic fields. The particle always gyrates around the magnetic field at the local gyrofrequency . The local perpendicular gyration velocity is determined by the requirement that the magnetic moment be a constant of the motion. This, in turn, fixes the local gyroradius . The parallel velocity of the particle is determined by Eq. (85). Finally, the perpendicular drift velocity is the sum of the drift velocity and the first-order drift velocity