Remarks on forced equations of the double pendulum type

Tarantello, Gabriella

doi:10.1090/S0002-9947-1991-1049620-3

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.

References

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
A. Capozzi, D. Fortunato, and A. Salvatore, Periodic solutions of Lagrangian systems with bounded potential, J. Math. Anal. Appl. 124 (1987), no. 2, 482–494. MR 887004, DOI 10.1016/0022-247X(87)90009-6
Kung-Ching Chang, Yi Ming Long, and Eduard Zehnder, Forced oscillations for the triple pendulum, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 177–208. MR 1039344
G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 259–281. Analyse non linéaire (Perpignan, 1987). MR 1019117, DOI 10.1016/S0294-1449(17)30025-2
Da Jun Guo, Jing Xian Sun, and Gui Jie Qi, Some extensions of the mountain pass lemma, Differential Integral Equations 1 (1988), no. 3, 351–358. MR 929922
Helmut Hofer, A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem, J. London Math. Soc. (2) 31 (1985), no. 3, 566–570. MR 812787, DOI 10.1112/jlms/s2-31.3.566
J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52 (1984), no. 2, 264–287. MR 741271, DOI 10.1016/0022-0396(84)90180-3
Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
Paul H. Rabinowitz, On a class of functionals invariant under a $\textbf {Z}^n$ action, Trans. Amer. Math. Soc. 310 (1988), no. 1, 303–311. MR 965755, DOI 10.1090/S0002-9947-1988-0965755-5
Gabriella Tarantello, Multiple forced oscillations for the $N$-pendulum equation, Comm. Math. Phys. 132 (1990), no. 3, 499–517. MR 1069833
Gabriella Tarantello, On the number of solutions for the forced pendulum equation, J. Differential Equations 80 (1989), no. 1, 79–93. MR 1003251, DOI 10.1016/0022-0396(89)90096-X