Description of Van Allen's Belt and Magnetic Mirrors The "Van Allen Belt" is a region near earth that concentrates a
high density of energy particles. The formation of this region originates in a phenomenon
called a magnetic mirror, when electrically charged particles are trapped in a
variable magnetic field region.
Given that, I want to describe the properties of this phenomenon from equations of electromagnetism, taking the "Van Allen Belt" as an object of study. If anyone can point me to some directions, I'll be grateful.
 A: First, look at the answers at: https://physics.stackexchange.com/a/670591/59023 and https://physics.stackexchange.com/a/671056/59023.
The existence of the radiation belts results from the quasi-stability of three single-particle motions in a dipole magnetic field in a plasma:  gyration, bounce, and drift.  I have listed these in order of fastest to slowest, i.e., the particles gyrate much faster than they bounce between the magnetic poles and much much faster than they drift around the Earth.
Gyration describes a particles cyclotron motion about a magnetic field and is only dependent upon the magnetic field strength and the particle's Lorentz factor (i.e., $\Omega_{cs} = \tfrac{ e \ B }{ \gamma \ m_{s} }$, where $e$ is the fundamental charge [C], $\gamma$ is the Lorentz factor [N/A], $B$ is the magnitude of the magnetic field [T], $m_{s}$ is the mass of species $s$ [kg]).
Bounce motion is one of the adiabatic invariants so long as the change in magnetic field strength and direction is slow compared to the bounce time between the poles.  If this assumption does not hold, a particle can be lost to the atmosphere (called precipitation).  Particles that are lost are said to be in the loss-cone – a range of pitch-angles (i.e., angle between particle momentum vector and the magnetic field vector) where particles are precipitated or escape from a magnetic bottle.
Finally, drift refers to the so called gradient drift velocity of particles in spatially varying magnetic field (e.g., see answer at https://physics.stackexchange.com/a/556682/59023).  So long as the gradient scale length is large compared to the particle's gyroradius, the particles will happily gradient drift around the Earth.

The formation of this region originates in a phenomenon called a magnetic mirror, when electrically charged particles are trapped in a variable magnetic field region.

This is kind of correct.  The formation of the radiation belts is much more complicated, obviously, but the basic idea is that particles can become trapped in these regions so long as the assumptions list above hold.  Most of the time there are two quasi-stable belts, an inner and outer belt.  The inner belt is almost entirely comprised of ions (mostly protons) and each particle has statistical lifetimes on the order of years or longer.  The outer belt is almost entirely comprised of electrons.  Under perfectly quiet and stable solar wind conditions, the lifetimes of these outer belt electrons can be weeks.  Under realistic solar wind conditions, the lifetimes are days or less.
As for how the radiation belts form, I have some detailed notes/comments at https://physics.stackexchange.com/a/490737/59023 and https://physics.stackexchange.com/a/142922/59023.
