我們簡短地闡述有擾動的克卜勒問題與其歷史上的發展，回顧與其相關的一些基本概念，並簡單估算在給定不同形式的位能時，無擾動克卜勒問題所得出的解。
接著我們引入K.C.Chen-K.J.Hsu的工作裡發現的二階微分不等式。透過反覆使用該不等式，使得我們得以簡單估算有擾動的克卜勒問題，並藉此證明當給予擾動向量一定程度的限制，有擾動的克卜勒問題與無擾動的克卜勒問題在碰撞點附近會有相似的漸進行為。
最後，我們就先前討論過的位能再做延伸，試圖討論更一般或更難以嚴格歸類的位能形式。

In this thesis, we provide a brief exposition of the perturbed Kepler problem and its historical development, and review some basic concepts associated with it. We also provide simple estimations of the solutions obtained for the unperturbed Kepler problem for different forms of potential.
Next, we introduce the second-order differential inequality discovered in the work of K.-C. Chen and K.-J. Hsu. By frequently applying this inequality, we are able to estimate the perturbed Kepler problem in a straightforward manner and thereby demonstrate that, with certain restrictions on the perturbations, the perturbed Kepler problem exhibits similar asymptotic behavior to the unperturbed Kepler problem near the collision points.
Finally, we extend our discussion to explore more general forms of potential, or forms that are harder to classify.

[1] V. Barutello, D.L. Ferrario, Susanna Terracini. On the singularities of gener-
alized solutions to n-body-type problems. International mathematics research
notices 2008.9, rnn069-rnn069, 2008.
[2] K.-C. Chen, K.-J. Hsu. The collision singularity of the Kepler problem with
singular perturbations. Proceedings of the American Mathematical Society(to
appear).
[3] F.N. Diacu. Regularization of partial collisions in the N-body problem. Dif-
ferential integral equations 5, 103-136, 1992.
[4] D.L. Ferrario, S. Terracini. On the existence of collisionless equivariant min-
imizers for the classical n-body problem. Inventiones mathematicae 155.2,
305-362, 2004.
[5] H.A. Pollard, D.G. Saari. Singularities of the n-body problem. I. Archive for
Rational Mechanics and Analysis 30.4, 263-269, 1968.
[6] H.J. Sperling. On the real singularities of the N-body problem. Journal f ̈ur
die reine und angewandte Mathematik 245, 15-40, 1970.
[7] H.J. Sperling. The collision singularity in a perturbed two-body problem.
Celestial Mechanics 1, 213–221, 1969.
[8] K.F. Sundman. M ́emoire sur le probl ́eme des trois corps. Acta mathematica
36.1, 105-179, 1913.
[9] A. Wintner. The analytical foundations of celestial mechanics. Courier Cor-
poration, 2014.