Given a smooth function
U
(
t
,
x
)
U(t,x)
,
T
T
-periodic in the first variable and satisfying
U
(
t
,
x
)
=
O
(
|
x
|
α
)
U(t,x) = \mathcal {O}(\vert x \vert ^{\alpha })
for some
α
∈
(
0
,
2
)
\alpha \in (0,2)
as
|
x
|
→
∞
\vert x \vert \to \infty
, we prove that the forced Kepler problem
x
¨
=
−
x
|
x
|
3
+
∇
x
U
(
t
,
x
)
,
x
∈
R
2
,
\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
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.