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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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An asymptotic universal focal decomposition for non-isochronous potentials
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by C. A. A. de Carvalho, M. M. Peixoto, D. Pinheiro and A. A. Pinto PDF
Trans. Amer. Math. Soc. 366 (2014), 2227-2263 Request permission


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:
  • M. M. Peixoto
  • Affiliation: Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
  • Email:
  • D. Pinheiro
  • Affiliation: Department of Mathematics, Brooklyn College of the City University of New York, Brooklyn, New York 11210
  • Email:
  • A. A. Pinto
  • Affiliation: LIAAD - INESC TEC and Department of Mathematics, Faculty of Science, University of Porto, Portugal
  • MR Author ID: 319558
  • Email:
  • Received by editor(s): October 20, 2011
  • Received by editor(s) in revised form: September 2, 2012
  • Published electronically: November 25, 2013
  • © Copyright 2013 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 2227-2263
  • MSC (2010): Primary 37E20, 34B15, 70H03, 70H09
  • DOI:
  • MathSciNet review: 3152729