An alternate Hamiltonian formulation of fourth–order theories and its application to cosmology

An alternate Hamiltonian H different from Ostrogradski’s one is found for the Lagrangian L = L(q, q̇, q̈), where ∂2L/∂(q̈)2 6= 0. We add a suitable divergence to L and insert a = q an d b = q̈. Contrary to other approaches no constraint is needed because ä = 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 6= 0 the Lagrangians L and L̂ = F̂ (R̂) with F̂ = 2R/G3 − 3L/G4, ĝij = G2 gij and R̂ = 3R/G2 − 4L/G3 give conformally equivalent fourth–order field equations being dual to each other. This generalizes Buchdahl’s result for L = R2. 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.