Variational Principle for Eigenvalue Problems of Hamiltonian Systems
Rafael D. Benguria, M. C. Depassier · Physical Review Letters · 1996
We consider the bifurcation problem ${u}^{\ensuremath{'}\ensuremath{'}}+\ensuremath{\lambda}u\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\ensuremath{\lambda}$. We give a variational characterization of the bifurcating branch $\ensuremath{\lambda}$ as a function of the 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.