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:

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

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:

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

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

\label{eq:alpha}
\frac{B_{0}}{B_{m}}=\frac{\sin^{2} \alpha_{0} }{\sin^{2} 90^{o} }=\sin^{2} \alpha_{0}

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

\label{eq:loss}
\sin^{2} \alpha_{LC}= \frac{B_{X}}{B_{a}}

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:

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

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:

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

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.