| Abstract: | Abstract Variational principles in which the Lagrangian is a scalar density and a function of a metric tensor and a vector field, together with their first derivatives, are investigated in a 4-dimensional space. Associated with such Lagrangians are two expressions, the metric Euler-Lagrange expression and the vector Euler-Lagrange expression. The most general Lagrangians (of this kind) for which either of these Euler-Lagrange expressions vanishes identically, are obtained. The most general Lagrangian (of this kind) for which the vector Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although this Lagrangian is more general than the one commonly used, it still has essentially the same energy-momentum tensor. The most general Lagrangian (of this kind) for which the metric Euler-Lagrange expression is precisely the electromagnetic energy-momentum tensor is derived. Although this Lagrangian is also more general than the one commonly used, the associated vector Euler-Lagrange equations are still Maxwell's equations. Finally it is shown that, in contrast to the situation which obtains in the case of scalar densities which are functions of up to second derivatives of the metric and first derivatives of the vector field, there does not exist a Lagrangian, of the kind under investigation, for which the metric Euler-Lagrange expression is precisely the Einstein tensor and the vector Euler-Lagrange expression vanishes identically. |
