Adiabatic invariants of alpha particles in the presence of time dependent perturbations

Theoretical and numerical analysis is done for the fast particles in tokamaks in an analogy with the nonlinear pendulum with slowly changing string length (A.Einstein, Solvay Conference, 1912). The two important parameters are the ratio between mode frequency and bounce frequency $\omega / \omega_b$ (change in string length versus pendulum frequency), as well as the perturbation amplitudes. Toroidal drift eigenmode structure is applied to the analysis.\footnote{C.Z.Cheng, Phys. Fluids {\bf 25}, 1020 (1982).}