  The purely affine, metric-affine and purely metric formulation of general
relativity are dynamically equivalent and the relation between them is
analogous to the Legendre relation between the Lagrangian and Hamiltonian
dynamics. We show that one cannot construct a dynamically equivalent, purely
affine Lagrangian from a metric-affine or metric F(R) Lagrangian, nonlinear in
the curvature scalar. Thus the equivalence between the purely affine picture
and the two other formulations does not hold for metric-affine and metric
theories of gravity with a nonlinear dependence on the curvature, i.e. F(R)
gravity does not have a purely affine formulation. We also show that this
equivalence is restored if the metric tensor is conformally transformed from
the Jordan to the Einstein frame, in which F(R) gravity turns into general
relativity with a scalar field. This peculiar behavior of general relativity,
among relativistic theories of gravitation, with respect to purely affine,
metric-affine and purely metric variation could indicate the physicality of the
Einstein frame. On the other hand, it could explain why this theory cannot
interpolate among phenomenological behaviors at different scales.
