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In a space of 4-dimensions, I will examine constrained variational problemsin which the Lagrangian, and constraint scalar density, are concomitants of a(pseudo-Riemannian) metric tensor and its first two derivatives. The Lagrangemultiplier for these constrained extremal problems will be a scalar field. Forsuitable choices of the Lagrangian, and constraint, we can obtainEuler-Lagrange equations which are second order in the scalar field and thirdorder in the metric tensor. The effect of disformal transformations on theconstraint Lagrangians, and their generalizations, is examined. This will yieldother second order scalar-tensor Lagrangians which yield field equations whichare at most of third order. No attempt is made to construct all possible thirdorder scalar-tensor Euler-Lagrange equations in a 4-space, although nineclasses of such field equations are presented. Two of these classes admitsubclasses which yield conformally invariant field equations. A few remarks onscalar-tensor-connection theories are also presented.

Lagrange Multipliers and Third Order Scalar-Tensor Field Theories