D ec 1 99 7 Conformal relations and Hamiltonian formulation of fourth – order gravity ∗

The conformal equivalence of fourth–order gravity following from a non–linear Lagrangian L(R) to theories of other types is widely known, here we report on a new conformal equivalence of these theories to theories of the same type but with different Lagrangian. For a quantization of fourth-order theories one needs a Hamiltonian formulation of them. One of the possibilities to do so goes back to Ostrogradski in 1850. Here we present another possibility: A Hamiltonian H different from Ostrogradski’s one is discussed 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 and 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. Finally, we discuss the stability properties of cosmological models within fourth–order gravity. PACS numbers: 04.50 Other theories of gravitation, 98.80 Cosmology, 03.20 Classical mechanics of discrete systems ∗Extended version of a lecture read at the International School-Seminar “Problems of Theoretical Cosmology”, 1-7 September 1997, Ulyanovsk, Russia