A slowly varying Hamiltonian with one degree of freedom and nearly closed orbits has an adiabatic invariant. This adiabatic invariant is conserved to all orders in ε, the slowness parameter, except for orbits that cross a separatrix. The present work discusses the change of the adiabatic invariant during this crossing process through order ε. First, a calculation of the change of the adiabatic invariant is presented. This calculation is general and, hence, encompasses previous results for specific cases. The change in the adiabatic invariant is shown to depend on a maximum of five parameters, which are functions of the Hamiltonian of interest. Second, the statistics of this process are derived. Finally, these results are applied to the motion of a particle in a wave with changing amplitude and phase velocity.

The reversed field pinch (RFP) is a configuration for plasma magnetic confinement. It has been traditionally viewed as dominated by a bath of MHD instabilities producing magnetic chaos and high energy transport. We report experimental results which go beyond this view. They show a decrease of magnetic chaos and the formation of a coherent helical structure in the plasma, whose imaging and temperature profile are provided for the first time. These quasi-single-helicity states are observed both transiently and in stationary conditions. The last case is consistent with a theoretically predicted bifurcation. Our results set a new frame for improving confinement in high current nonchaotic RFP's.

When a parameter in the Hamiltonian of a one-degree-of-freedom oscillator is slowly varied at rate $\epsilon$, an adiabatic invariant exists which is conserved to all orders in $\epsilon$, except on phase-space orbits which cross a separatrix. In the present work, the change in the adiabatic invariant due to a separatrix crossing is given to order $\epsilon$ for a wide class of Hamiltonian systems. This result is applied to the special case of a charged particle moving under the influence of an electrostatic wave with slowly varying amplitude and frequency.