Variation of the radial and vertical actions as a function of integration time for axisymmetric potentials. The actions are computed at each point along the orbit displayed on the right in Figure 10. The actions are calculated using three different approximations: (a) the adiabatic approximation actionAngleAdiabatic, (b) the Stäckel approximation actionAngleStaeckel, and (c) a general orbit-integration-based method actionAngleIsochroneApprox. The radial and vertical actions for this orbit are 0.05 and 0.009, respectively. The Stäckel approximation conserves the radial action ≈5 times better than the adiabatic approximation and the vertical action ≈10 times better. The actionAngleIsochroneApprox with its default settings performs about 20 and 5 times better than the Stäckel approximation. All methods agree on the actions to within these errors.