A Variational Principle for Eigenvalue Problems of Hamiltonian Systems

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