  An alternate Hamiltonian H different from Ostrogradski's one is found for the
Lagrangian L = L(q, \dot q, \ddot q). We add a suitable divergence to L and
insert a=q and b=\ddot q. Contrary to other approaches no constraint is needed
because \ddot a = b is one of the canonical equations. Another canonical
equation becomes equivalent to the fourth-order Euler-Lagrange equation of L.
Usually, H becomes quadratic in the momenta, whereas the Ostrogradski approach
has Hamiltonians always linear in the momenta. For non-linear L=F(R), G=dF/dR
\ne 0 the Lagrangians L and \hat L=\hat F(\hat R) with \hat F=2R/G\sp 3-3L/G\sp
4, \hat g_{ij}=G\sp 2g_{ij} and \hat R=3R/G\sp 2 - 4L/G \sp 3 give conformally
equivalent fourth-order field equations being dual to each other. This
generalizes Buchdahl's result for L=R^2. The exact fourth-order gravity
cosmological solutions found by Accioly and Chimento are interpreted from the
viewpoint of the instability of fourth-order theories and how they transform
under this duality. Finally, the alternate Hamiltonian is applied to deduce the
Wheeler-De Witt equation for fourth-order gravity models more systematically
than before.
