Mathematical Methods and N-body Problem in Celestial Mechanics

Dear Colleagues,

The N-body problem is a classical problem in celestial mechanics, dealing with predicting the motion of a system of N celestial bodies under the influence of gravitational forces. Newton’s formulation of the laws of motion and the law of universal gravitation revolutionized the understanding of celestial mechanics. By applying his laws of motion and gravitation to the Sun and a planet, Newton was able to derive Kepler's three laws, which describe the orbits of planets around the Sun. More than two hundred years later, Poincaré proved that a limiting case of the three-body problem is unsolvable in a certain technical sense: it admits ‘homoclinic tangles’ and therefore is not integrable by analytic functions. Poincare's work into the three-body problem led to the discovery of the chaos theory. The N-body problem is essential for understanding the motion of planets, moons, stars, and galaxies within the universe. Many research groups worldwide continue to dedicate their efforts to studying various aspects of the problem, such as the analysis of central configurations or the determination of periodic solutions and choreographies. Additionally, they focus on studying some of its special configurations, such as the N-body ring problem, the Ollongren problem, the Caledonian problem, and many others.

The N-body problem entails solving a set of differential equations describing the positions and velocities of each body over time. Traditional methods for tackling this problem include numerical integration techniques such as Euler's method, Runge–Kutta methods, and adaptive step-size methods. However, the N-body problem poses significant computational challenges, particularly for systems with large numbers of bodies or complex interactions. In recent years, researchers have turned to artificial intelligence (AI) and neural networks to address these challenges and develop alternative approaches for solving the N-body problem. Artificial intelligence techniques offer innovative solutions for modeling and predicting the behavior of complex dynamical systems like the N-body problem. Machine learning algorithms, in particular, excel at discovering patterns and making predictions from data. In the context of the N-body problem, AI methods can learn from observations of celestial body motions to infer underlying dynamical laws, thereby bypassing the need for explicit mathematical models. Neural networks, a fundamental component of artificial intelligence, have shown promise in simulating and predicting the behavior of N-body systems. By training neural networks on large datasets of simulated or observed celestial motions, researchers can develop models which are capable of accurately predicting the future trajectories of bodies within the system. These neural network models can capture complex interactions and nonlinear dynamics that may be challenging to represent using traditional mathematical approaches. Hybrid approaches combining traditional mathematical methods with artificial intelligence techniques offer a powerful framework for tackling the N-body problem. For instance, researchers may use numerical integration methods to generate training data for neural networks, which can then refine and improve upon the initial predictions. Conversely, neural networks can assist in accelerating numerical simulations by providing fast and accurate approximations of complex interactions.

Dr. Juan F. Navarro
Guest Editor

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• N-body ring problem
• N-body problem
• three-body problem
• hill problem
• stability
• periodic orbits
• coorbital satellites
• planetary rings
• AI
• numerical methods for the N-body problem
• hybrid methods
• neural networks