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

TL;DR

This work advances the search for symmetric periodic solutions in the $n$-body problem by combining two scalable optimisation strategies, MP-AIDEA and Lattice, with discretisations of the action functional (via points or Fourier coefficients) on symmetry-reduced fundamental domains. It develops a robust framework to identify multiple critical points, assess their stability through the discrete Morse index and level-distance metrics, and construct non-minimal solutions using a regularised Mountain Pass approach. The methodology yields concrete examples for symmetry groups $\mathbb{Z}_2$ and $D_6$, including isolated and non-isolated minima, and demonstrates the practical viability of retrieving a richer set of periodic orbits in the symmetric $n$-body problem. Overall, the approach provides a principled, scalable pipeline for uncovering complex orbital structures with potential implications for dynamical systems and celestial mechanics research.

Abstract

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.

Figures (14)

• Figure 1: Visualisation of a two-dimensional example of the Lattice algorithm. The first figure displays the set $M$ along with the distributions of the variables. The subsequent images illustrate the progressive division of the domain at each layer, following the process described in Example \ref{['Example lattice']}.
• Figure 2: Lattice Scheme
• Figure 3: Circular orbit of action level $9.802$
• Figure 4: Ducati orbit of action level of $10.442$
• Figure 5: Orbit of action level $13.579$

Theorems & Definitions (17)

• Example 2.1
• Remark
• Remark
• Definition 4.1: Computational discrete Morse index
• Definition 4.2: Intra and trans level distances
• Definition 5.1: Path
• Theorem 5.2: Theorem 1.4, BT04
• Corollary 5.3: Corollary 1.5, BT04
• Corollary 5.4: Corollary 1.6, BT04
• Definition 5.5: Palais-Smale Condition