An exact magnetic-moment invariant of charged-particle gyromotion

For the motion of a charged particle in a uniform, time-dependent axial magnetic field B(t)ez, it is shown that there is an exact magnetic-moment invariant of the particle dynamics M, to which the adiabatic magnetic-moment invariant μ=mv ⊥ 2/2B is asymptotic when the time scale of the magnetic field variation is much slower than the gyroperiod. The connection between the exact invariant M and the adiabatic invariant μ enables us to characterize in detail the robustness of the adiabatic magnetic-moment invariant μ.