First adiabatic invariant

The first adiabatic invariant, $μ$ , is the action integral over a cyclotron frequency. It is usually written:

$\mu=\frac{mv_\perp^2}{2B}$

[edit] Relation to Action Integral

The action integral is written:

$\oint p\,dq=\oint_{\mathrm{gyro}}m v_\perp\,ds$

We can define the length as $d s = ρ d θ$ , and use the gyro radius $\rho=v_\perp mc/qB$ :

$\oint p\,dq=\oint_0^{2\pi}\frac{m^2 c v_\perp^2}{qB}\,d\theta$

Over one orbit, if we are adiabatic then B and $v_\perp$ change much more slowly than the gyro period $Ω c$ . This gives:

$\oint p\,dq=\frac{2\pi m^2 c v_\perp^2}{qB}=\frac{4\pi mc}{q}\mu$