标题： 通过变分方法和优化技术获得的对称n体问题的新解 显示英文标题

标题： New solutions for the symmetrical n-body problem through variational approach and optimisation techniques

摘要： 变分方法在$n$体问题上的进展在天体力学领域取得了显著的进步，揭示了可能轨道的新类型。 本文中，使用进化算法分析了与$n$体问题相关的拉格朗日作用量的临界点，以识别离散系统的周期性和对称性解。 一个关键目标是定位作用泛函的最小点，因为这些点对应于满足系统微分方程的可行周期解。 通过采用随机和确定性算法，我们探索了解空间，并获得了这些轨道的数值表示。 接下来，我们将这些轨道视为临界点来检查它们的稳定性。 一种方法是计算它们的离散莫尔斯指数，以区分最小点和鞍点。 另一种方法是根据它们的作用水平对其进行分类。 最后，通过分析吸引盆地的边界，我们可以利用 安布罗塞蒂-拉宾诺维奇山口定理识别非最小临界点。 这导致该算法的一个更新版本，提供了定理的构造性证明，在特定情况下产生新的轨道。 本文建立并扩展了\cite{nostro}中提出的结果，提供了更详细的理论框架和对公式的深入见解。 此外，我们展示了新的数值结果，并对发现的临界点进行了扩展分析，进一步增强了先前研究的成果。 显示英文摘要

摘要： Advances in the variational approach to the $n$-body problem have led to significant progress in celestial mechanics, uncovering new types of possible orbits. In this paper, critical points of the Lagrangian action associated with the $n$-body problem are analysed using evolutionary algorithms to identify periodic and symmetrical solutions of the discretised system. A key objective is to locate minimum points of the action functional, as these correspond to feasible periodic solutions that satisfy the system's differential equations. By employing both stochastic and deterministic algorithms, we explore the solution space and obtain numerical representations of these orbits. Next, we examine the stability of these orbits by treating them as critical points. One approach is to compute their discrete Morse index to distinguish between minimum points and saddle points. Another is to classify them based on their action levels. Finally, analysing the boundaries of their attraction basins allows us to identify non-minimal critical points via the Ambrosetti-Rabinowitz Mountain Pass Theorem. This leads to an updated version of the algorithm that provides a constructive proof of the theorem, yielding new orbits in specific cases. This paper builds upon and extends the results presented in \cite{nostro}, providing a more detailed theoretical framework and deeper insights into the formulation. Additionally, we present new numerical results and an extended analysis of the critical points found, further enhancing the findings of the previous study.