Add an adiabatic invariants demo

[henry2004y]

• Perform numerical experiments and compare with theoretical expectations for the 3 adiabatic invariants.
• The 1st adiabtic invariant can be checked by calculating the perpendicular velocity and local B field along the trajectory.
• The bounce period in a dipole field can be approximately expressed as (Hamlin+, 1974)

$$ T_b = \frac{4\pi R_E (1.30 - 0.56 \sin\alpha_{eq})}{\sqrt{2W/m}} $$

where $\alpha_{eq}$ is the equatorial pitch angle with the relation

$$ \sin^2\alpha_{eq} = \frac{B_{eq}}{B_m} = \frac{\cos^6\lambda_m}{(1+3\sin^2\lambda_m)^{1/2}} $$

where $B_{eq}$ is the magnetic field strength at the equator, $B_m$ is the magnetic field strength at the mirror point, and $\lambda_m$ is the magnetic latitude of the mirror point.

• From the 3rd adiabatic invariant in a dipole field, we can derive an expression for the drift period. This can be checked numerically.

We want to create a demo to demonstrate the three periods corresponding to the 3 adiabatic invariants.