Periodic solutions to a forced Kepler problem in the plane

Boscaggin, Alberto; Dambrosio, Walter; Papini, Duccio

doi:10.1090/proc/14719

Periodic solutions to a forced Kepler problem in the plane
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by Alberto Boscaggin, Walter Dambrosio and Duccio Papini

Proc. Amer. Math. Soc. 148 (2020), 301-314

DOI: https://doi.org/10.1090/proc/14719

Published electronically: July 30, 2019

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Abstract:

Given a smooth function $U(t,x)$, $T$-periodic in the first variable and satisfying $U(t,x) = \mathcal {O}(\vert x \vert ^{\alpha })$ for some $\alpha \in (0,2)$ as $\vert x \vert \to \infty$, we prove that the forced Kepler problem \begin{equation*} \ddot x = - \dfrac {x}{|x|^3} + \nabla _x U(t,x),\qquad x\in \mathbb {R}^2, \end{equation*} has a generalized $T$-periodic solution, according to the definition given in the paper by A. Boscaggin, R. Ortega, and L. Zhao [Trans. Amer. Math. Soc. 372 (2019), 677–703]. The proof relies on variational arguments.

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