Fundamentals of Plasma Physics 2025, 21 - Adiabatic Invariants
Assume a frame moving with the velocity orthogonal to the magnetic field. The inductive electric field emerges from the product of Lorentz equation and velocity vector. The time-average over the Larmor orbit of the the products of perpendicular components of velocity and electric field describes the rate at which the perpendicular electric field does work on the particle. From this, it becomes apparent that the first adiabatic invariant μ = W⊥/B is a conserved quantity. It's equal to the ratio of kinetic energy of gyromotion and gyrofrequency.
It follows that the magnetic moment of current loops are also adiabatically conserved, as is the magnetic flux enclosed by a gyro-orbit. The conservation of μ is equivalent to a constant angular momentum due to axisymmetry.
The magnetic field strength varies along the direction of the field, so the field must be curved, with varying density of field lines. At infinity, the field lines are squeezed together to ensure a vanishing gradient for the magnetic field. The periodicity of this creates potential hills and valleys, acting as "magnetic mirrors". At W0 < μBmax, the particles bounce between the mirrors of the magnetic well. Otherwise they are impeded, but not reflected. From the μ-conservation, the reflection condition is actually only dependent on the ratio of minimal and maximal B-field density, which can be expressed as a trapping angle for the particles.
The second adiabatic invariant is defined I = ∮P||ds = ∮(mv|| + qA||)ds. It's invariant if all time-dependence of the magnetic well is slow compared to bounce frequencies of trapped particles, and any spatial inhomogeneities of the well field are so gradual the bounces only change minorly between bounces. Where μ-invariance is related to the perpendicular components, I-invariance is related to the parallel ones.