OPTIMIZATION OF INTERMEDIATE GRAVITATIONAL FORCES IN THE RESTRICTED THREE-BODY PROBLEM

Аннотация

The restricted three-body problem is a classical and complex challenge in celestial mechanics, involving the motion of a small body under the gravitational influence of two massive primaries. This paper focuses on optimizing the intermediate gravitational forces acting within this system to enhance computational efficiency and trajectory stability. By employing advanced numerical integration techniques and modern optimization algorithms such as genetic algorithms and particle swarm optimization, we explore strategies for minimizing energy expenditure and improving predictive accuracy in orbit calculations. Applications in spacecraft trajectory planning, orbital insertion, and mission design are also discussed. The study contributes to ongoing efforts to model and control chaotic dynamical systems in astrodynamics with greater precision. Keywords: Restricted three-body problem, gravitational optimization, numerical integration, orbital mechanics, celestial dynamics, trajectory planning, chaotic systems, genetic algorithms, particle swarm optimization, space mission design. 1. Introduction The restricted three-body problem (RTBP) is a fundamental and historically significant problem in classical mechanics and astrodynamics. It describes the motion of a small, massless body—such as a spacecraft—under the gravitational influence of two massive primary bodies (e.g., the Earth and the Moon), which themselves follow Keplerian orbits around their common center of mass. Unlike the two-body problem, which has an exact analytical solution, the RTBP exhibits nonlinear, often chaotic behavior that makes it analytically intractable in the general case. The RTBP serves as a simplified yet powerful model for studying real-world space missions, particularly those involving libration point orbits, interplanetary trajectories, and multi-body gravitational interactions. One of the critical challenges in this problem is optimizing the intermediate gravitational forces to achieve stable or efficient orbits. Accurate modeling of these forces enables better prediction of trajectories, fuel-efficient maneuvers, and reliable mission planning in complex gravitational environments. Recent advancements in computational methods and optimization algorithms have made it possible to revisit the RTBP with renewed focus. Techniques such as https://scientific-jl.com/wsrj 3 7 5 Volume-40_Issue-1_June-2025 World scientific research journal Runge-Kutta integration, symplectic methods, and modern heuristic optimization algorithms (e.g., genetic algorithms and particle swarm optimization) offer promising tools to address the problem's inherent sensitivity to initial conditions and nonlinearity. This paper aims to explore how such methods can be applied to optimize gravitational interactions within the RTBP framework. We discuss the mathematical formulation of the system, present key numerical methods for trajectory simulation, evaluate optimization strategies, and highlight their practical applications in space exploration. 2. Literature Review The restricted three-body problem (RTBP) has been extensively studied since the work of Euler, Lagrange, and Poincaré. Despite its apparent simplicity, the RTBP exhibits complex and often chaotic dynamics that have challenged mathematicians, physicists, and engineers for centuries. 2.1 Classical Foundations Initial contributions to the RTBP came from Joseph-Louis Lagrange, who identified the five equilibrium points (now called Lagrange points), which remain central to mission planning and orbit design. Henri Poincaré later showed that the RTBP does not possess a general analytical solution, laying the groundwork for modern chaos theory and qualitative analysis of dynamical systems. 2.2 Numerical Integration Techniques With the advancement of computational resources, numerical methods have become essential for simulating the RTBP. Runge-Kutta methods are widely used for their balance between accuracy and computational efficiency. More recent work emphasizes symplectic integrators, which preserve the system's Hamiltonian structure and are particularly valuable for long-term orbital simulations. Researchers such as Hairer et al. (2006) and Sanz-Serna (1994) have explored the application of structure-preserving algorithms to better handle the sensitivity of the RTBP to initial conditions, thereby improving the reliability of long-term integrations. 2.3 Optimization in Celestial Mechanics Optimization techniques in the RTBP context aim to find optimal trajectories, control inputs, or mission parameters. Genetic algorithms (GAs) and particle swarm optimization (PSO) have gained attention for solving nonlinear, high- dimensional problems where gradient-based methods struggle. Miele (2003) and Conway (2010) have demonstrated the effectiveness of these algorithms in trajectory optimization and mission design. Further studies by Betts (1998) introduced direct collocation methods combined with nonlinear programming so