The problem of constructing variational principles for a given second-order quasi-linear partial differential equation is considered. In particular, we address the problem of finding a first-order function f whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian. Two sets of equations for such a function f are obtained. Necessary and sufficient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set. In certain special cases, such as the case when the operator is elliptic, the problem is completely solved. The utility of our results is illustrated by a variety of examples.