Adiabatic Invariants

Poincaré invariants are generally of little practical interest unless the curve $C$ closely corresponds to the trajectories of actual particles. Now, for the motion of magnetized particles it is clear from Eqs. (60) and (73) that points having the same guiding centre at a certain time will continue to have approximately the same guiding centre at a later time. An approximate Poincaré invariant may thus be obtained by choosing the curve $C$ to be a circle of points corresponding to a gyrophase period. In other words,

$${\cal I} \simeq I = \oint {\bf p}\cdot \frac{\partial{\bf q}}{\partial\gamma}\, d\gamma.$$ (106)

Here, $I$ is an adiabatic invariant.

To evaluate $I$ for a magnetized plasma recall that the canonical momentum for charged particles is

$${\bf p} = m\,{\bf v} +e\,{\bf A},$$ (107)

where ${\bf A}$ is the vector potential. We express ${\bf A}$ in terms of its Taylor series about the guiding centre position:

$${\bf A}({\bf r}) = {\bf A}({\bf R}) + (\mbox{\boldmath$\rho$}\cdot\nabla)\,{\bf A}({\bf R}) + O(\rho^2).$$ (108)

The element of length along the curve $C(t)$ is [see Eq. (74)]

$$d{\bf r} = \frac{\partial\mbox{\boldmath$\rho$}}{\partial \gamma}\,d\gamma = \frac{{\bf u}}{{\Omega}}\,\,d\gamma.$$ (109)

The adiabatic invariant is thus

$$I = \oint \frac{{\bf u}}{{\Omega}} \cdot \left\{ m\,({\bf U}... ...}\cdot \nabla)\,{\bf A}\right]\right\}\,d\gamma + O(\epsilon),$$ (110)

which reduces to

$$I = 2\pi\,m\,\frac{u_\perp^{~2}}{{\Omega}} + 2\pi\,\frac{e}{... ...x{\boldmath$\rho$}\cdot \nabla)\,{\bf A}\rangle + O(\epsilon).$$ (111)

The final term on the right-hand side is written [see Eq. (76)]

$$2\pi\,e\,\langle (\mbox{\boldmath$\rho$}\times{\bf b}) \cdot... ... \nabla\times{\bf A} = -\pi\,m\,\frac{u_\perp^{~2}}{{\Omega}}.$$ (112)

It follows that

$$I = 2\pi\, \frac{m}{e}\,\mu + O(\epsilon).$$ (113)

Thus, to lowest order the adiabatic invariant is proportional to the magnetic moment $\mu$.