The first kind of motion that a charged particle has in a magnetic field is a gyration around the field lines. If the magnetic field is \(\overrightarrow{B}\) the frequency of gyration is \(\omega_{B}=\frac{eB}{m}\) and the gyroradius is \(r_{B}=\frac{mv_{\perp}}{Be}\) where \(v_{\perp}\) is the component of the velocity perpendicular to the magnetic field and \(v_{\Vert}\) is parallel. If the only force acting on the particle is the magnetic one, no work is done on the particle and the

magnetic flux \(\phi_{m}\) is constant, so:

\begin{equation}\label{eq:fluxmag}

\phi_{m}=B \pi r^{2}_{B}=\frac{2\pi m E^{\perp}_{k}}{e^{2}B}

\end{equation}

where \(E^{\perp}_{k}\) is the kinetic energy associated with the transverse motion and so is \(\frac{mv^{2}_{\perp}}{2}\). From eq. \ref{eq:fluxmag} we have \(\frac{E^{\perp}_{k}}{B}=constant\) so the magnetic moment of the current loop \(\mu\) given by:

\begin{equation}\label{eq:mu}

\mu = i\times A = \frac{ev_{\perp}}{2 \pi r_{B}}\pi r^{2}_{B} = \frac{E^{\perp}_{k}}{B}

\end{equation}

is constant too (and is called {\it first adiabatic invariant}). This invariant is related to the bounce motion (motion parallel to the field lines). We have calling {\it pitch angle} \(\alpha\) the angle bewtween perpendicular and parallel velocity component (so \(\tan \alpha = \frac{v_{\perp}}{v_{\Vert}}\)) \(\frac{E_{k}\sin ^{2} \alpha}{B}=constant\). Because the ratio \(\frac{E^{\perp}_{k}}{B}\) is constant, when a particle moves toward the Earth magnetic pole, \(v_{\perp}\) and \(\sin^{2}\alpha\) must increase (the magnetic field \(B\) increases), eventually reaching the value \(\alpha=90^{o}\). In this point the particle motion is reversed and this is called the {\it mirror point} (that correspond to a value \(B_{m}\) or mirror field). This mechanisn approximate the {\it magnetic bottle} mechanism in a non-uniform field. The total energy of the particle is not changing, so \(v\) is constant and calling \(B_{0}\) and \(\alpha_{0}\) the values of the magnetic field and the pitch angle at the geomagnetic equator, we can write the relation

\begin{equation}\label{eq:alpha}

\frac{B_{0}}{B_{m}}=\frac{\sin^{2} \alpha_{0} }{\sin^{2} 90^{o} }=\sin^{2} \alpha_{0}

\end{equation}

If particles encounter the atmosphere before they bounce back they will be lost by absorption (interaction) with it, this happens for all particles that in any location \(X\) have a picth angle \(\alpha _{LC}\) that is \(\alpha < \alpha_{LC}\), where

\begin{equation}\label{eq:loss}

\sin^{2} \alpha_{LC}= \frac{B_{X}}{B_{a}}

\end{equation}

where \(B_{X}\) is the intensity in \(X\) and \(B_{a}\) is the intensity at the intersection of the field line with the atmosphere. There are two more adiabatic invariants, the first (namely the {\it second adiabatic invariant}) one is related to the effect called the {\it curvature-gradient drift}. This effect is due to both the gradient of the field approaching the Earth and the curvature of the field lines that both produce a drift motion in the plane perpendicular to the dipole axis. So electrons are driven eastward and protons westwards. The second adiabatic invariant is expressed as:

\begin{equation}\label{eq:second}

J_{2} = p \int^{A}_{A'} \sqrt{1-\frac{B}{B_{m}} dl_{b}}

\end{equation}

or better \(I_{}=\frac{J_{2}}{2p} \simeq constant\).

The {\it third adiabatic invariant} is related to a slowly changing magnetic field, where first and second adiabatic invariants (usually conserved in a static magnetic field) are still conserved, the particle momentum may change, but the magnetic flux in a drift path is conserved. This is:

\begin{equation}\label{eq:third}

J_{3} = \int (q \overrightarrow{A}+ \overrightarrow{p})d\overrightarrow{l}_{d} \simeq q\phi \simeq constant

\end{equation}

where \(\phi\) is the magnetic flux enclosed in the drift path. The invariance is violated if changes occur in a time \(t< \tau_{d}\), where \(\tau_{d}\) is the drift period.

Magnetic trapping occurs if the respective periods have a strict jerarchy, so \(\tau_{g} < \tau_{b}< \tau_{d}\), where \(\tau_{g}\) is gyration period and \(\tau_{b}\) is the bouncing period.