Solutions of three-body problem based on an equivalent system approach
Solutions of three-body problem based on an equivalent system approach
Generalised reduced masses with a set of equations governing the three relative motions between two of 3-bodies in their gravitational field are established, of which the dynamic characteristics of 3-body dynamics, fundamental bases of this paper, are revealed. Based on these findings, an equivalent system is developed, which is a 2-body system with its total mass, constant angular momentum, kinetic and potential energies same as the total ones of three relative motions, so that it can be solved using the well-known theory of 2-body system. From the solution of equivalent system with the revealed characteristics of three relative motions, the general theoretical solutions of 3-body system are obtained in the curve-integration forms along the orbits in the imaged radial motion space. The possible periodical orbits with generalised Kepler’s law are presented. Following the description and mathematical demonstrations of the proposed methods, the examples including the Euler’s / Lagrange’s problems, and a reported numerical one are solved to valid the proposed methods. The methods derived from 3-body system are extended to N-body problems.
Three-body problem, Equivalent system with solutions, Orbit-equation of a conic section, Generalised reduced mass, Chaotic motions, Generalised Kepler’s law, N-body problem.
Xing, Jing T
d4fe7ae0-2668-422a-8d89-9e66527835ce
17 January 2026
Xing, Jing T
d4fe7ae0-2668-422a-8d89-9e66527835ce
Xing, Jing T
(2026)
Solutions of three-body problem based on an equivalent system approach.
Acta Mechanica Sinica, 42 (1), [524942].
(doi:10.1007/s10409-025-24942-x).
Abstract
Generalised reduced masses with a set of equations governing the three relative motions between two of 3-bodies in their gravitational field are established, of which the dynamic characteristics of 3-body dynamics, fundamental bases of this paper, are revealed. Based on these findings, an equivalent system is developed, which is a 2-body system with its total mass, constant angular momentum, kinetic and potential energies same as the total ones of three relative motions, so that it can be solved using the well-known theory of 2-body system. From the solution of equivalent system with the revealed characteristics of three relative motions, the general theoretical solutions of 3-body system are obtained in the curve-integration forms along the orbits in the imaged radial motion space. The possible periodical orbits with generalised Kepler’s law are presented. Following the description and mathematical demonstrations of the proposed methods, the examples including the Euler’s / Lagrange’s problems, and a reported numerical one are solved to valid the proposed methods. The methods derived from 3-body system are extended to N-body problems.
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Accepted/In Press date: 20 May 2025
e-pub ahead of print date: 17 January 2026
Published date: 17 January 2026
Keywords:
Three-body problem, Equivalent system with solutions, Orbit-equation of a conic section, Generalised reduced mass, Chaotic motions, Generalised Kepler’s law, N-body problem.
Identifiers
Local EPrints ID: 509494
URI: http://eprints.soton.ac.uk/id/eprint/509494
ISSN: 0567-7718
PURE UUID: 40f74040-4a3b-4340-925a-183cb411c080
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Date deposited: 24 Feb 2026 17:46
Last modified: 07 Mar 2026 02:42
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