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The collisionless equation in the title transports a gyrophase-independent phase-space distribution function at fixed first adiabatic invariant. Its leading guiding center position has $\dot{\mathbf{R}}={\mathbf{u}}_{\perp }+v_{\parallel }\mathbf{b}$. The parallel acceleration includes $(q/m)E_{\parallel }-\mu \mathbf{b}\cdot \nabla B$ and, when required, terms from motion of the local frame. Both $v_{\parallel }$ and the parallel derivative refer to the exact perturbed magnetic field.

The slow dynamics of charged-particle guiding centers after averaging over rapid cyclotron motion, assuming gyroradii much smaller than resolved perpendicular length scales. Depending on the ordering, the retained transverse motion may include just the common electric drift or also magnetic curvature drift, gradient-B drift and polarization drift. These choices give distinct limits, not interchangeable closures.

The transverse guiding center drift produced by a magnetic field strength gradient, with first adiabatic invariant $\mu =v_{\perp }^2/(2B)$ and signed cyclotron frequency ${\Omega }_s$. Together with magnetic curvature drift, it produces velocity-dependent transverse transport.

The guiding center is the center of a charged particle's circular cyclotron orbit. In a nonuniform field it drifts slowly compared with the rapid cyclotron frequency.

Rapid circular particle motion perpendicular to a locally uniform magnetic field, superposed on parallel streaming when the parallel velocity is nonzero. In Gaussian units, its signed cyclotron frequency is given in the title and its gyroradius is $v_{\perp }/|{\Omega }_s|$. Slowly varying fields permit a guiding center description.

The transverse guiding center drift associated with magnetic field-line curvature. The signed cyclotron frequency ${\Omega }_s=q_{s}B/(m_{s}c)$ makes oppositely charged species drift oppositely at fixed velocity. The resulting species-dependent response is retained in slow drift-scale instabilities.

A charge $q$ moving nonrelativistically in prescribed fields obeys the Lorentz force lawThe magnetic part is perpendicular to velocity, so $d(m{v}^2/2)/dt=q\mathbf{E}\cdot \mathbf{v}$: the electric field can change kinetic energy, while the magnetic field alone changes direction without doing work.

In a uniform electric field with no magnetic field, acceleration is constant: $\mathbf{v}={\mathbf{v}}_0+(q\mathbf{E}/m)t$ and $\mathbf{r}={\mathbf{r}}_0+{\mathbf{v}}_{0}t+q\mathbf{E}{t}^2/(2m)$. The trajectory is a parabola when there is initial motion perpendicular to the field, and a line when motion is purely along it. Reversing the charge reverses the electric acceleration.

In a uniform magnetic field with no electric field, resolve velocity into parallel and perpendicular components. The parallel component is constant; the perpendicular component rotates at the cyclotron frequencyThe resulting path is a helix, or a circle if $v_{\parallel }=0$, with pitch $2\pi v_{\parallel }/{\omega }_c$. Charge sign determines its sense of rotation. These relations underlie magnetic bending and mass/charge analysis. At relativistic energies, use momentum $\mathbf{p}=\gamma m\mathbf{v}$; at fixed energy the cyclotron frequency becomes $|q|B/(\gamma m)$.

For uniform crossed fields $\mathbf{E}\perp \mathbf{B}$, the velocity ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/B^2$ satisfies $\mathbf{E}+{\mathbf{v}}_D\times \mathbf{B}=0$. The orbit is a gyromotion about a guiding center drifting withindependent of mass and charge sign. A velocity selector transmits particles with the appropriate direction and speed $E/B$ without deflection. If the electric field has a component along the magnetic field, that component accelerates the parallel motion instead. Nonuniform fields can add other drifts; the simple helix and uniform crossed-field expressions require the stated uniform-field assumptions.