论文标题
扰动开普勒问题的正规变分原理
Regularized variational principles for the perturbed Kepler problem
论文作者
论文摘要
本文的目的是开发一种方法，该方法将使用变异技术与正则化方法结合使用，以研究与扰动的Kepler System \ [\ ddot x = - \ \ \ \ frac {x} {x | x | x | x | x | x | x | x | x | x | x | x | t）的生存和多样性结果和dirichlet问题，并\]其中$ d \ geq 1 $，而$ p：\ mathbb {r} \ to \ mathbb {r}^d $很平滑，$ t $ - periodic，$ t> 0 $。 与问题相关的动作功能的关键点存在通过非本地变化的变化，这是由Levi-Civita和Kustaanheimo-Stiefel Techniques启发的变量。作为一个应用程序，我们将证明，受扰动的开普勒问题对$ d = 2 $和$ d = 3 $的$ T $周期解决方案无限为$ p $。
The goal of the paper is to develop a method that will combine the use of variational techniques with regularization methods in order to study existence and multiplicity results for the periodic and the Dirichlet problem associated to the perturbed Kepler system \[ \ddot x = -\frac{x}{|x|^3} + p(t), \quad x \in \mathbb{R}^d, \] where $d\geq 1$, and $p:\mathbb{R}\to\mathbb{R}^d$ is smooth and $T$-periodic, $T>0$. The existence of critical points for the action functional associated to the problem is proved via a non-local change of variables inspired by Levi-Civita and Kustaanheimo-Stiefel techniques. As an application we will prove that the perturbed Kepler problem has infinitely many generalized $T$-periodic solutions for $d=2$ and $d=3$, without any symmetry assumptions on $p$.
