Variational principle for eigenvalue problems of Hamiltonian systems

We consider the bifurcation problem u '' + lambda u = N(u) with two point boundary conditions where N(LI) is a general nonlinear term which may also depend on the eigenvalue lambda. We give a variational characterization of the bifurcating branch lambda as a function of th amplitude of the solution. As an application we show how it can be used to obtain simple approximate closed formulas for the period of large amplitude oscillations.