We consider the eigenvalues, \(\lambda_ n(\rho)\), of self-adjoint Sturm- Liouville systems to be real valued functionals of certain coefficient functions in the differential equation. We introduce a classical (in general nonlinear) functional K(\(\rho)\) which is tangent to \(\lambda_ n(\rho)\) at a fixed function \(\rho^*\). That is, \(\lambda_ n(\rho^*)=K(\rho^*)\) and \(\delta \lambda_ n=\delta K\) at \(\rho^*\). Then by using classical calculus of variations on K(\(\rho)\) we show how to find extremals of \(\lambda_ n(\rho)\) over certain classes of functions \(\rho\).