Exact solution to the problem of N bodies forming a multi-layer rotating structure

Area of Science:

• Celestial Mechanics
• Gravitational Dynamics
• N-body Problem

Background:

• Investigates Newtonian gravitational interaction for N material points in concentric circular orbits.
• Structures involve N^2 orbits with N^3 axisymmetrically located, identical mass bodies rotating around a symmetry axis.
• These configurations are known as homographic-dynamics or planar central configurations.

Purpose of the Study:

• To find exact solutions for the dynamics of complex, rotating N-body gravitational systems.
• To analyze the forces and motion equations within these symmetrical structures.
• To develop computational tools for calculating and visualizing these configurations.

Main Methods:

• Employs Hamiltonian mechanics for systems with fewer than 20 bodies.
• Derives differential motion equations, reducing them to linear algebraic equations for body masses.
• Utilizes a developed computer program (RtCrcSt2.for) for parameter specification and characteristic calculations.

Main Results:

• Identified forces acting on each body within the rotating structure.
• Obtained solutions in various forms, enabling calculations for structures up to one million bodies.
• Presented graphical representations and described properties of the calculated structures.

Conclusions:

• The study provides exact solutions for complex, symmetrical gravitational systems.
• Discusses stability issues and potential applications in celestial and space mechanics.
• Demonstrates the scalability of the method to very large N-body systems.