Vertical vs. radial action over five orbital periods along the orbit displayed on the right in Figure 10, calculated with the axisymmetric action–angle methods. The top panels shows this for actions calculated using the adiabatic approximation as well as for its grid-based-interpolation version. The bottom panel shows the actions computed with the Stäckel method and its grid-based-interpolation version. The direct Stäckel actions are also shown as black points in the top panel to emphasize that the actions are much better conserved using the Stäckel method. Because the adiabatic actions only depend on the approximate integrals of the motion that are used to tabulate the actions for interpolation, the interpolated values agree exactly with the direct calculation (to within the interpolation errors; the crosses and dots almost exactly overlap). As the tabulation of the Stäckel method uses approximate integrals that are not depended on in the direct calculation, the grid-based values deviate from the direct values. In particular, the grid-based J_z varies by about a factor of three more than the direct J_z.