Remarks on forced equations of the double pendulum type
Author:
Gabriella Tarantello
Journal:
Trans. Amer. Math. Soc. 326 (1991), 441452
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 periodic in each of the variables . We study periodic solutions for the corresponding equation of motion subject to a periodic force . If has mean value zero, the corresponding variational problem admits a symmetry which yields distinct periodic solutions (see [9]). Here we consider the case where the average of , 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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 A. Capozzi, D. Fortunato, and A. Salvatore, Periodic solutions of Lagrangian systems with bounded potential, J. Math. Ann. Appl. 124 (1987), 482494. MR 887004 (88h:58026)
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 K. C. Chang, Y. Long, and E. Zehnder, Forced oscillations for the triple pendulum, Preprint. MR 1039344 (91h:58021)
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 G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 259282. MR 1019117 (91b:58034)
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 D. Guo, J. Sun, and G. Qi, Some extension of the mountainpass lemma, Differential Integral Equations 1 (1988), 351358. MR 929922 (89e:58024)
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 P. Rabinowitz, On a class of functionals invariant under a action, Trans. Amer. Math. Soc. 310 (1988), 303311. MR 965755 (89i:34057)
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 G. Tarantello, Multiple forced oscillations for the pendulum equation, Comm. Math. Phys. 132 (1990), 499517. MR 1069833 (91j:34060)
 [11]
 G. Tarantello, On the number of solutions for the forced pendulum equation, J. Differential Equations 80 (1989), 7993. MR 1003251 (90h:34058)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110496203
PII:
S 00029947(1991)10496203
Article copyright:
© Copyright 1991
American Mathematical Society
