Adiabatic invariants

For periodic motions the theory of mechanics shows that quantities called actions can remain invariant for slow changes in the system. An action J can be defined in terms of generalized coordinates q_gen and conjugate momenta p_gen by

$$J = \oint p_{gen} d q_{gen} \ ,$$ (2.31)

where the integral is over one period of the motion. A particle's gyromotion is one example of a periodic motion amenable to the construction of an approximate constant of the motion (or invariant). Defining q_gen to be the gyrophase $\phi$, then the angular momentum $p_{gen} = l = m v_{\perp} r_{L}$ is the conjugate momentum. Inserting these variables into (2.31) and integrating one finds

$$J = 2\pi \frac{m}{q} \frac{v_{\perp}^{2}}{B} = \frac{4\pi m}{q} \mu$$ (2.32)

for slowly varying B. The quantity

$$\mu = \frac{v_{\perp}^{2}}{2 B}$$ (2.33)

is known as the first adiabatic invariant of a plasma particle. This implies that a particle's perpendicular energy $W_{\perp} = 1/2 m v_{\perp}^{2}$ is proportional to B if $\mu$ is constant.

Another, perhaps more obvious derivation of the first adiabatic invariant is as follows. Assume that the particle sees a small change in ${\bf B}$ during a gyroperiod, whether due to temporal or spatial variations in ${\bf B}$. I.E.,

$$\frac{1}{\Omega_{c}} \vert \frac{\partial B}{\partial t} B^{-1} \vert \ll 1 \ .$$ (2.34)

The change in $W_{\perp}$ in one gyroperiod is

$$\Delta W_{\perp} q \oint {\bf E} . {\bf d}l$$ = (2.35)

$$\frac{1}{c} \frac{\partial }{\partial t} \left( q \int_{S} {\bf B} . d{\bf S} \right) \ .$$ = (2.36)

Assuming the orbit size changes very little in one gyroperiod then

$$\Delta W_{\perp} \approx q \pi r_{L}^{2} \frac{1}{c} \frac{\partial B}{\partial t} .$$ (2.37)

Since the change in B in one gyroperiod is

$$\Delta B = \frac{2\pi}{\Omega_{c}} \frac{\partial B}{\partial t}$$ (2.38)

then $\Delta W_{\perp} = W_{\perp} \Delta B / B$ or

$$\Delta \left( \frac{W}{B} \right) = 0 = \Delta( \mu )\ .$$ (2.39)

That is, $\mu$ is a constant.

Other adiabatic invariants also exist. The second or longitudinal adiabatic invariant is associated with the periodic bouncing of particles in magnetic flux tubes and magnetic bottles:

$$J_{L} = m \oint v_{\parallel} d z$$ (2.40)

A third adiabatic invariant can be associated with the periodic drift of a particle (due to $\nabla B$ and curvature drifts) around a dipole magnetic field. It is useful for studying particle motions in Earth's magnetosphere but is not addressed further here.