  A generalized Hamiltonian definition of ghost surfaces (surfaces defined by
an action-gradient flow) is given and specialized to the usual Lagrangian
definition. Numerical calculations show uncorrected quadratic-flux-minimizing
(QFMin) and Lagrangian ghost surfaces give very similar results for a chaotic
magnetic field weakly perturbed from an integrable case in action-angle
coordinates, described by $L = L_0 + \epsilon L_1$, where $L_0(\dot{\theta})$
(with $\dot{\theta}$ denoting $d\theta/d\zeta$) is an integrable field-line
Lagrangian and $\epsilon$ is a perturbation parameter. This is explained using
a perturbative construction of the auxiliary poloidal angle $\Theta$ that
corrects QFMin surfaces so they are also ghost surfaces. The difference between
the corrected and uncorrected surfaces is $O(\epsilon^2)$, explaining the
observed smallness of this difference. An alternative definition of ghost
surfaces is also introduced, based on an action-gradient flow in $\Theta$,
which appears to have superior properties when unified with QFMin surfaces.
