This talk will outline the analysis of critical points of a parametrized functional that arises in the study of linear elliptic eigenvalue problems. These critical points describe the eigenvalues and eigenfunctions of the elliptic boundary value problem. The functional has well-defined second derivatives and an associated Morse index is defined that characterizes which eigenvalue is associated with the critical point. The results are exemplified for an unconstrained variational principle for Steklov eigenproblems of the Laplacian. This is joint work with Professor Giles Auchmuty.