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

Trans. Amer. Math. Soc. 326 (1991), 441-452

DOI: https://doi.org/10.1090/S0002-9947-1991-1049620-3

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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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