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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An asymptotic universal focal decomposition for non-isochronous potentials


Authors: C. A. A. de Carvalho, M. M. Peixoto, D. Pinheiro and A. A. Pinto
Journal: Trans. Amer. Math. Soc. 366 (2014), 2227-2263
MSC (2010): Primary 37E20, 34B15, 70H03, 70H09
Published electronically: November 25, 2013
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Abstract: Galileo, in the seventeenth century, observed that the small oscillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equilibrium. It is well known that, for small oscillations of the pendulum and small intervals of time, the dynamics of the pendulum can be approximated by the dynamics of the harmonic oscillator. We study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected. We analyze such dynamical behaviour through a renormalization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we obtain a universal asymptotic focal decomposition for this family of mechanical systems. This paper is intended to be the first in a series of articles aiming at a semiclassical quantization of systems of the pendulum type as a natural application of the focal decomposition associated to the two-point boundary value problem.


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Additional Information

C. A. A. de Carvalho
Affiliation: Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
Email: aragao@if.ufrj.br

M. M. Peixoto
Affiliation: Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Email: peixoto@impa.br

D. Pinheiro
Affiliation: Department of Mathematics, Brooklyn College of the City University of New York, Brooklyn, New York 11210
Email: dpinheiro@brooklyn.cuny.edu

A. A. Pinto
Affiliation: LIAAD - INESC TEC and Department of Mathematics, Faculty of Science, University of Porto, Portugal
Email: aapinto@fc.up.pt

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05995-8
PII: S 0002-9947(2013)05995-8
Keywords: Mechanical systems, renormalization, universality, focal decomposition
Received by editor(s): October 20, 2011
Received by editor(s) in revised form: September 2, 2012
Published electronically: November 25, 2013
Article copyright: © Copyright 2013 American Mathematical Society