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

 

Remarks on forced equations of the double pendulum type


Author: Gabriella Tarantello
Journal: Trans. Amer. Math. Soc. 326 (1991), 441-452
MSC: Primary 58F22; Secondary 34C25, 58E05, 58F05, 70H35, 70K40
MathSciNet review: 1049620
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1049620-3
PII: S 0002-9947(1991)1049620-3
Article copyright: © Copyright 1991 American Mathematical Society