  In this paper, we establish an equivalence between force-free fields and
conformally geodesic fields, and between harmonic fields and conformally
eikonal fields in the context of conformal geometry. In contrast to previous
work, our approach and equivalence results generalize to arbitrary dimensions.
In accordance with three-dimensional theory, our defining equations emerge as
the Euler-Lagrange equations of hierarchies of variational principles -
distinguished by the topological constraints they impose - and retain the known
inclusions of the special cases from each other. Specifically, we relate
stationary points of hierarchies of $L^2$ resp. $L^1$-optimization problems by
a conformal change of metric, provide an explicit construction of the conformal
factors relating the relevant metrics and identify the field lines of physical
vector fields fields as conformal geodesics. Despite the allowed topological
complexity of the fields under consideration, these observations reveal
geometric order which is obtained by merely pointwise rescaling of the metric.
