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Phase space derivation of a variational principle for one-dimensional Hamiltonian systems

Affiliation:	1. Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata-700032, India;2. SISSA, Via Bonomea 265, 34136 Trieste, Italy;3. INFN, Sezione di Trieste, Italy;4. IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy;5. Erwin Schrödinger International Institute for Mathematical Physics, University of Vienna, Vienna, Austria;6. Isfahan University of Technology, Isfahan 84156-83111, Iran;7. Perimeter Institute for Theoretical Physics, 31 Caroline St. N, N2L 2Y5, Waterloo ON, Canada;1. UNESP - Campus de Guaratinguetá - DFQ, Avenida Dr. Ariberto Pereira da Cunha 333, CEP 12516-410, Guaratinguetá, SP, Brazil;2. IFQ - Universidade Federal de Itajubá, Av. BPS 1303, Pinheirinho, Caixa Postal 50, 37500-903, Itajubá, MG, Brazil;3. Instituto de Ciência e Tecnologia, Universidade Federal de Alfenas Rod. José Aurélio Vilela (BR 267), Km 533, n°11999, CEP 37701-970, Poços de Caldas, MG, Brazil;4. Brazil;1. School of Physics and Astronomy, Cardiff University, Queen’s Buildings, The Parade, Cardiff, CF24 3AA, UK;2. School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK;3. School of Physics, HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK

Abstract:	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. A new derivation of a variational principle for the lowest eigenvalue λ is given. This derivation makes use only of simple algebraic inequalities and leads directly to a more explicit expression for the eigenvalue than what had been given previously.

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