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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Remarks on forced equations of the double pendulum type
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by Gabriella Tarantello PDF
Trans. Amer. Math. Soc. 326 (1991), 441-452 Request permission

Abstract:

Motivated by the double pendulum equation we consider Lagrangian systems with potential $V = V(t,q)$ periodic in each of the variables $t,q = ({q_1}, \ldots ,{q_N})$. We study periodic solutions for the corresponding equation of motion subject to a periodic force $f = f(t)$. If $f$ has mean value zero, the corresponding variational problem admits a ${{\mathbf {Z}}^N}$ symmetry which yields $N + 1$ distinct periodic solutions (see [9]). Here we consider the case where the average of $f$, though bounded, is no longer required to be zero. We show how this situation becomes more delicate, and in general it is only possible to claim no more than two periodic solutions.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 441-452
  • MSC: Primary 58F22; Secondary 34C25, 58E05, 58F05, 70H35, 70K40
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1049620-3
  • MathSciNet review: 1049620