Fourier Transformation of Solutions to Three‑Body and N‑Body Problems: From Newtonian Closed‑Form Solutions to Einsteinian Geometric Spectra

Author: Zhang Suhang

    Luoyang School of Mathematics

Abstract

Classical many‑body dynamics has long been trapped by the obsession with analytic closed‑form solutions. Newton required physical solutions to be finite, elementary and closed formulas. While the two‑body problem satisfies this criterion, Poincaré proved that the three‑body system admits no such global solution, giving rise to the widespread misinterpretation that “the three‑body problem has no solution”. This paper proposes a solution‑characterization framework for many‑body systems based on Fourier‑series theory, which transforms the trajectories of three‑body and N‑body systems into multi‑circle superposition on the complex plane. The Fourier functional forms for two‑body, three‑body and N‑body systems are presented. It is demonstrated that solutions to the three‑body problem do exist; they merely transit from “finite elementary closed forms” to “infinite frequency‑domain geometric spectra”. Fourier‑series solutions retain the determinism and geometric attributes of Newtonian mechanics and respond to Einstein’s pursuit of determinism and geometric unification for physical systems. Taking the Sun‑Earth‑Moon system as an example, geometric configurations from two‑circle and three‑circle superposition are illustrated. This study shows that graphical representations constitute a valid form of solution. Insisting on obtaining two‑body‑style point‑wise analytic formulas for the three‑body problem contradicts the diversity of natural laws.

Keywords: three‑body problem; many‑body dynamics; Fourier series; closed‑form solution; geometric spectrum; determinism

1 Introduction: Newton’s Obsession with Closed‑Form Solutions and Poincaré’s Discovery

Since Newton’s era, classical mechanics has adopted an inherent aesthetic criterion: a genuine physical solution must be a finite, elementary, closed analytic formula. The great success of the two‑body problem — orbits are conic sections and positions can be directly computed via r = p/(1+e\cos\theta) — elevated this criterion to a paradigm.

When researchers attempted to extend this paradigm to the three‑body problem, they encountered fundamental obstacles. Poincaré proved that no global elementary closed‑form solution valid over the entire time domain exists for the three‑body system. Nevertheless, this result is frequently misread as “the three‑body problem has no solution”.

This is not the case. This paper argues that solutions to the three‑body problem objectively exist; their form merely transits from “finite single‑circle formulas” to “infinite multi‑circle spectra”. This transition does not overturn the deterministic framework of Newtonian mechanics. Instead, it conforms at a higher level to Einstein’s ideas regarding the geometric nature and determinism of the physical world.

2 Mathematical Foundation: Fourier Multi‑Circle Superposition

2.1 Single‑Circle Motion

Uniform circular motion on the complex plane is written as:

z(t) = R e^{i\omega t}

where R denotes amplitude and \omega denotes angular frequency.

2.2 Fourier Series

Any periodic function z(t) with period T=2\pi/\omega_0 can be expanded into a Fourier series:

z(t) = \sum_{k=-\infty}^{+\infty} c_k e^{ik\omega_0 t}

The coefficients c_k are uniquely determined by initial conditions.

Geometric interpretation: any periodic motion is equivalent to the vector superposition of infinitely many uniform circular motions.

2.3 Quasi‑Periodic Motion

When a system contains multiple incommensurable frequencies \omega_1,\omega_2,\dots,\omega_N (their ratios are not all rational numbers), the motion is quasi‑periodic and can be expressed by a multi‑frequency Fourier series:

z(t) = \sum_{k_1,\dots,k_N} c_{k_1\dots k_N} e^{i(k_1\omega_1+\dots+k_N\omega_N)t}

This serves as the general characterization for three‑body and many‑body systems.

3 From Two‑Body to Three‑Body: Fourier Characterization

3.1 Two‑Body System (Two‑Circle Superposition)

Taking the Sun as reference, the relative motion of Earth in the two‑body approximation can be approximated by superposition of two frequency components:

z_{2}(t) = R_1 e^{i\omega_1 t} + R_2 e^{i\omega_2 t}

‑ R_1,\omega_1: amplitude and angular frequency of the reference circle;

‑ R_2,\omega_2: amplitude and angular frequency of the rotating circle.

Geometric outcome: the composite trajectory forms a regular periodic curve corresponding to the classical elliptical two‑body orbit.

3.2 Three‑Body System (Three‑Circle Superposition)

The Sun‑Earth‑Moon system is represented by superposition of three rotating vectors:

z_{3}(t) = R_1 e^{i\omega_1 t} + R_2 e^{i\omega_2 t} + R_3 e^{i\omega_3 t}

‑ R_3,\omega_3: rotational amplitude and angular frequency corresponding to the Moon.

Geometric outcome: the composite trajectory exhibits multi‑layer nesting and complex morphology. Since \omega_1,\omega_2,\omega_3 are generally incommensurable, the trajectory displays quasi‑periodic or chaotic behaviour.

3.3 N‑Body System (N‑Circle Superposition)

Under quasi‑periodic approximation, an N‑body system reads:

z_N(t) = \sum_{k=1}^{N} R_k e^{i\omega_k t}

N=2 corresponds to the two‑body case; N=3 corresponds to the three‑body case.

Note: This formula describes vector superposition with fixed frequencies. Under strong coupling or chaotic dynamics, the angular frequencies of components are no longer constant, so orbits cannot be fully described by a set of invariant frequencies.

4 Transformation toward the Newtonian Form

4.1 Newton’s Requirements

Newton’s criteria for closed‑form solutions consist of three points:

1. Finite terms: infinite series are excluded;

2. Elementary functions: only arithmetic operations, roots, trigonometric, exponential and logarithmic functions are permitted;

3. Closed form: substituting given initial conditions yields direct evaluation of physical quantities.

The two‑body problem satisfies all three criteria, whereas the three‑body problem does not.

4.2 What is Preserved in Fourier‑Series Solutions

Although Fourier series are infinite expansions, they retain two core traits of Newtonian mechanics:

 Newtonian two‑body closed‑form solution Fourier spectral solution 

Finite terms Yes No 

Elementary functions Yes No 

Determinism Yes Yes 

Geometric nature Yes Yes 

Crucially, every term R_k e^{i\omega_k t} in the Fourier series represents a geometric circle. Given initial conditions, all coefficients R_k,\omega_k are uniquely determined with no randomness.

The three‑body problem is not unsolvable; its solution is the natural extension of the Newtonian two‑body geometric solution into the infinite‑dimensional frequency domain.

5 Einstein’s Geometry of Determinism

5.1 “God does not play dice”

Einstein insisted on fundamental determinism of the physical world.

The Fourier multi‑circle superposition model indicates that chaos in three‑body orbits does not originate from intrinsic randomness in underlying physics. Instead, it arises from complex behaviour generated by superposition of infinitely many deterministic circular motions. Even though the governing equations are deterministic, chaotic systems are extremely sensitive to initial values, rendering long‑term orbit prediction practically infeasible. Given initial conditions, Fourier spectral coefficients are fully determined, and the position at any moment is strictly governed by the series. This reflects the deterministic viewpoint.

5.2 Geometrization of Physics

Einstein devoted his career to the geometrization of physics, as realised in general relativity.

The Fourier representation on the complex plane:
z(t) = \sum_{k=1}^{\infty} R_k e^{i\omega_k t}
Each term corresponds to one geometric object (a circle), and the overall motion is the superposition of these geometric objects. Three‑body motion amounts to geometric weaving in the frequency domain.

5.3 Unification of Fields and Waves

Fourier series constitute fundamental tools for field theory and wave analysis. Transforming discrete celestial motion into frequency‑domain continuous spectra essentially elevates the many‑body problem to a field‑theoretic description. While Einstein pursued a unified field theory, Fourier geometric spectra provide an idea for frequency‑domain unified description of many‑body motion.

6 Graphical Solutions: Why Graphs Qualify as Solutions

6.1 Limitations of the Traditional Definition of “Solution”

Within conventional dynamical viewpoints, only analytic formulas count as “genuine solutions”. Numerical results and geometric images are treated merely as approximations or auxiliary tools.

6.2 Three‑Fold Validity of Graphical Solutions

表格
Dimension Argument
Mathematics Convergence of Fourier series guarantees that images are geometric realisations of the series
Physics Geometric solutions in phase space (pioneered by Poincaré) are already recognised
Information Graphs fully encode global configuration, hierarchical order and evolutionary boundaries of a system

6.3 Sun‑Earth‑Moon Example

表格
System Fourier form Graphical feature
Sun‑Earth Two‑circle superposition Regular periodic trajectory
Earth‑Moon Two‑circle superposition Secondary nested trajectory
Sun‑Earth‑Moon Three‑circle superposition Multi‑layer intertwined network

The composite trajectory graph of a three‑body system constitutes one complete type of solution for that system.

6.4 Innovations of This Study

The innovations of this research lie chiefly in methodological re‑interpretation. The Fourier series is a well‑established mathematical tool. The novelty of this paper is that multi‑rotating‑circle superposition is promoted from a conventional orbit‑fitting tool to a geometric‑spectrum characterization framework for many‑body dynamics. Within this framework, two‑body, three‑body and N‑body motions are unified under the same multi‑circle‑vector picture. Meanwhile, geometric graphical configurations are established as valid forms of solution. This enables a re‑examination of the popular misinterpretation regarding the “unsolvable three‑body problem”, and connects Newton’s closed‑solution paradigm, Poincaré’s topological ideas and Einstein’s programme for physical geometrization.

This does not mean that the three‑body problem has been completely solved; rather, it opens a new representational approach. Instead of persistently searching for a finite elementary closed‑form expression, we adopt the geometric‑spectrum framework of Fourier multi‑circle superposition to describe system motion. By recognizing orbital geometric configurations as a valid form of solution, we preserve the determinism and geometric features of dynamics while breaking free from the cognitive limitations imposed by Newton’s closed‑solution paradigm.

Identifying why this framework cannot fully satisfy Newton’s ideal closed‑form criteria also constitutes a contribution of the present work. Three underlying reasons may be distinguished. First, the three‑body system exhibits intrinsic chaos; tiny deviations grow exponentially over time, so no fixed finite set of elementary trigonometric components can rigorously describe orbital evolution for all time. Second, Newton conflated practical computation with theoretical judging criteria. Though he accepted truncations and approximations in astronomical calculations, he imposed “finite elementary closed‑form together with rigorous global validity across all time” as a rigid yardstick for perfect solutions. Third, mathematically, Poincaré’s results demonstrate that no global elementary closed‑form solution exists for the three‑body problem, rendering this dual‑condition requirement intrinsically unrealisable. The geometric‑spectrum representation in this paper abandons these unattainable formal constraints while preserving the deterministic and geometric core of dynamics, offering a representational route applicable for finite‑time intervals under controllable precision.

7 Conclusions

1. Solutions are not unique in form. The two‑body problem admits closed analytic solutions, while the three‑body problem possesses Fourier spectral solutions, numerical solutions and topological solutions. Graphs represent a valid form of solution.
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2. Fourier multi‑circle superposition preserves dynamical determinism: chaos of three‑body orbits originates from superposition of infinitely many deterministic circular motions, with no probabilistic randomness at the fundamental level.
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3. From Newton to Einstein: Newton pursued finite elementary closed forms, whereas Einstein pursued physical determinism and geometrization. Fourier spectral solutions inherit both ideas, upgrading finite single‑circle solutions to geometric spectra composed of infinite circles.
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4. The three‑body problem is not unsolvable. Demanding a two‑body‑style analytic closed‑form formula for the three‑body problem represents an obsession contradicting the diversity of natural laws.

The order of the universe is diverse rather than minimally unified.