Quadratic Lagrangians and Palatini's device

This paper deals with the mutual inequivalence of g-variation and P-variation of a given action S, the components of linear connection GAMMA sup(m)sub(kl) being required to be symmetric. Under g-variation S is required to be stationery with respect to variations of the metric tensor gsub(ij), the GAMMA sup(m)sub(kl) being taken-to be Christoffel symbols from the outset, whereas under P-variations the gsub(ij) and GAMMAsup(m)sub(kl) are initially regarded as mutually independent and S is required to be stationery with respect to independent variations of these quantities. The discussion is illustrated at length by examples in which the Lagrangian of S is one or another of a set of homogeneous or inhomogeneous quadratic invariants of the Riemann tensor. (author)