摄动Kepler问题的正则化变分原理
Regularized variational principles for the perturbed Kepler problem
摘要：
本文的目标是开发一种方法，将结合使用变分技术与正则化方法，以研究存在性和多重性结果的周期性和Dirichlet问题相关的扰动开普勒系统\[ \ddot x = -\frac{x}{|x|^3} + p(t), \quad x \in \mathbb{R}^d, \]其中$d\geq 1$，和$p:\mathbb{R}\to\mathbb{R}^d$是顺利和$T$ -周期，$T>0$。受Levi-Civita和Kustaanheimo-Stiefel技巧的启发，通过变量的非局部变量变换证明了该问题作用泛函临界点的存在性.作为一个应用，我们将证明扰动开普勒问题对于$d=2$和$d=3$有无穷多个广义$T$ -周期解，而对$p$没有任何对称性假设.
英文摘要：
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 $p$.
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