Topological Classification and Snapshot Solution of Classical Three-Body and N-Body Problems Based on DOG-FCE Steady-State Constraints

Author: Zhang Suhang

Institution: Luoyang School of Mathematics

Abstract

The three-body and N-body problems remain core challenges in classical mechanics and computational astrophysics due to strong nonlinear coupling and chaos arising from initial-condition sensitivity. Traditional numerical methods rely on high-precision continuous time-stepping integration, which inevitably suffers from error accumulation and chaotic amplification during long-timescale evolution, making reliable determination of long-term topological outcomes difficult. Based on the Discrete Order Geometry (DOG) framework, this paper establishes a new topology-driven, non-time-stepping computational paradigm for many-body evolution.

Within this paradigm, each celestial body is defined as an independent spacetime lattice cell. Classical continuous physical laws are preserved inside each lattice, while dynamical coupling between lattices is encoded via the Zhang sparse matrix. The Fundamental Coupling Equation (FCE) in its steady-state form directly solves for the global topological snapshot of the system at a specified target time without full time integration. A classical chaotic three-body system is adopted as a validation example. The results show that the DOG-FCE model avoids error divergence inherent to continuous integration. Using the existence of steady-state solutions, it enables precise topological classification of long-term evolutionary outcomes of many-body systems, distinguishing bound steady configurations from topological-breakup ejection states. This discrete topological analysis framework provides an efficient, low-cost and robust new solution path for chaotic many-body systems.

Keywords: three-body problem; N-body dynamics; Discrete Order Geometry (DOG); FCE equation; topological snapshot; chaotic system; sparse coupling

1 Introduction

Since Poincaré proved that no general algebraic analytical solution exists for the three-body problem, it has been widely recognized that orbital evolution of strongly coupled many-body systems within the continuous differential framework exhibits irreducible sensitivity to initial conditions, hindering long-term precise orbital tracking.

Widely adopted numerical solvers, including Runge–Kutta schemes and symplectic integrators, are built upon the underlying logic of continuous spacetime differential equations plus sequential time-step integration. They carry three inherent limitations:

1. Process dependence: Intermediate evolutionary states must be computed sequentially; direct solving for the system state at a distant target time is not supported.

2. Chaotic accumulation: Small numerical round-off errors and minor deviations in initial conditions are amplified exponentially with integration steps, eventually corrupting orbital predictions.

3. Computational redundancy: For large star clusters and weakly coupled many-body systems, numerous celestial bodies with no meaningful dynamical interaction still participate in operations over dense global matrices, wasting computing resources.

To address these bottlenecks, this paper departs from conventional continuous evolution schemes and introduces the layered architecture of Discrete Order Geometry (DOG). A many-body dynamical model is constructed following the principle of continuity locally, discreteness globally. Instead of tracking chaotic intermediate orbits step by step, the work focuses on determining and solving for steady-state topological outcomes. The FCE steady-state constraint equation enables cross-time snapshot solving and topological classification of many-body systems, establishing an analytical paradigm distinct from classical celestial mechanics.

2 Construction of the DOG Lattice Dynamical Model

2.1 Definition of Celestial Spacetime Lattices

For a many-body system with N celestial bodies, each body is abstracted as an independent self-consistent spacetime lattice node carrying full dynamical parameters:

S_i = S_i(x, y, z, v_x, v_y, v_z, m, t)

The model adopts hierarchical physical rules:

- Inside a lattice: Newtonian gravitation, classical orbital mechanics and other continuous physical laws are fully retained, compatible with all well-established local dynamical results.

- Global scale across lattices: Superposition of a global continuous gravitational field is discarded. Dynamical connections between celestial bodies are implemented exclusively through discrete topological coupling.

2.2 The Zhang Sparse Coupling Matrix

Conventional N-body models use an N\times N dense coupling matrix with O(N^2) complexity, containing a large number of non-effective coupling terms.

This paper introduces the Zhang sparse coupling matrix, which accurately filters and truncates dynamical correlations among celestial bodies:

Z_{ij} = \begin{cases} C_{ij} & \text{if bodies } i,j \text{ have effective gravitational and kinematic coupling} \\ 0 & \text{if bodies } i,j \text{ have no significant dynamical correlation} \end{cases}

Here C_{pq} denotes the normalized coupling weight coefficient. This sparsification removes massive redundant calculations and drastically reduces the iteration complexity of large-scale many-body systems.

2.3 FCE Steady-State Constraint Equation for Snapshot Solving

The global topological stability of a many-body system is governed by the steady-state form of the Fundamental Coupling Equation (FCE):

\lambda_p \omega_p \prod_{q \in \text{Neigh}(p)} C_{pq} \omega_q = 1

Where: \omega_p is the orbital dynamical state variable of the celestial lattice; \lambda_p is the single-body steady-state correction coefficient; C_{pq} is the neighbor coupling weight taken from the Zhang matrix.

Core solution logic: For any given target time t^*, sequential time integration is unnecessary. One directly solves the above system of nonlinear algebraic equations. A convergent real solution corresponds to the topological steady-state snapshot of the system at t^*, characterizing the macroscopic configuration of the many-body system at that moment.

3 Topological Classification of Many-Body Systems Based on the FCE Solution Space

A central difficulty of chaotic three-body and many-body systems is forecasting long-term evolution. Relying on properties of the FCE solution space, this paper defines two rigorous topological classes to bypass uncertainty in chaotic processes.

3.1 Bound Steady State (Topological Conservation)

At target time t^*, if the FCE equation admits a valid real steady-state solution and all neighbor coupling weights remain active without broken topological connections, the system is in a topologically conserved state.

The many-body system forms a bound stable configuration, including periodic orbits, quasi-periodic orbits and Lagrangian steady configurations. No star escapes and no structural disintegration occurs.

3.2 Chaotic Tearing State (Topological Breakup)

If the coupling weight of a celestial body tends toward zero, effective topological links break, and the FCE steady-state constraint has no solution, topological breakup is identified.

Physically, this scenario corresponds to gravitational slingshot ejection of one body and disintegration of the original bound many-body structure. The typical outcome is a single ejected body and the remaining pair forming a stable binary system, consistent with classic evolutionary pathways of three-body systems.

Core proposition: Intermediate chaotic orbits cannot be predicted precisely over long timescales, yet long-term topological outcomes of the system can be judged accurately via the existence of FCE steady-state solutions.

4 Empirical Example Design and Paradigm Comparison

4.1 Example Setup

A classical perturbed chaotic three-body system is selected: three equal-mass bodies initialized in a weakly perturbed near-equilateral Lagrangian triangle configuration, a highly initial-condition-sensitive setup prone to chaotic divergence.

4.2 Control Experiment Scheme

Control group (traditional integration): Fourth-order Runge–Kutta (RK4) high-precision sequential integration with a small time step. Long-term evolution is fully simulated, tracking error accumulation and the moment of system breakup.

DOG-FCE experimental group: A 3\times3 Zhang sparse coupling matrix is constructed with initial orbital and mass parameters supplied. The target evolution duration is set directly, and the FCE steady-state topological snapshot at the final time is solved.

4.3 Characteristic Results of the Example

1. Computational efficiency gap: Long-duration RK4 integration requires extensive iteration and considerable runtime. The DOG model relies only on matrix operations and algebraic solving, achieving millisecond-scale computation with substantial efficiency gains.

​

2. Consistency of end states: Within the weakly perturbed quasi-static regime, topological classification, ejection judgment and remaining bound configurations obtained by DOG are highly consistent with end states from high-precision traditional integration.

​

3. Error mechanism advantage: With no sequential integration steps, cumulative numerical errors from long-time evolution are eliminated.

4.4 Model Scope of Applicability

The steady-state FCE model presented here does not resolve intermediate continuous orbital evolution. Its primary functions are judging long-term topological outcomes and solving steady-state configurations. It applies to weakly coupled, slowly evolving, quasi-static many-body systems and serves as a complementary paradigm to conventional integrators.

5 The New Computational Paradigm of Discrete Topological Astrophysical Dynamics

This work establishes a new research branch complementary to classical continuous celestial mechanics: Discrete Topological Astrophysical Dynamics. The two paradigms are clearly contrasted below:

1. Classical continuous paradigm: System of differential equations → iteration over small time steps → full process integration → final state accumulated along the evolution path; process-dependent and vulnerable to chaotic error growth.

​

2. DOG topological paradigm: Discrete spacetime lattices → sparse matrix encoding of couplings → FCE steady-state constraint solving → direct output of target topological snapshot; chaotic intermediate paths are bypassed and end states are prioritized.

This paradigm is broadly applicable to chaotic three-body systems, multiple-star moving groups, space debris cloud evolution, stability assessment of large stellar streams, and end-state analysis of black-hole multi-body systems, with strong theoretical extensibility and engineering value.

6 Conclusion

Poincaré’s classical result defines the non-integrability and chaotic sensitivity of orbital evolution for three-body systems under the continuous differential framework; it does not imply that topological outcomes of many-body systems are undecidable.

Through the layered architecture of Discrete Order Geometry, coupling reconstruction via the Zhang sparse matrix, and snapshot solving under FCE steady-state constraints, this paper shifts the focus of many-body dynamics from tracking chaotic intermediate orbits toward determining convergent topological end states.

The model demonstrates that evolutionary processes of complex many-body systems may be chaotic and unsuitable for precise stepwise forecasting, while bound steady states, topological breakup and stellar ejection outcomes can be solved and classified directly using purely algebraic steady-state constraints.

The establishment of Discrete Topological Astrophysical Dynamics provides a new, efficient and self-consistent mathematical language for chaotic many-body problems constrained by continuous integration frameworks for centuries, filling the paradigm gap in long-term topological forecasting within classical celestial mechanics.

References

 (Omitted)