Fourier Transformation of Three-body and Many-body Solutions: From Newtonian Closed-form Solutions to Einsteinian Geometric Spectra

Author: Zhang Suhang

Abstract

Classical many-body dynamics has long been trapped by the obsession with "analytic closed-form solutions". Newton postulated that a physical solution must be a finite, elementary, closed formula. The two-body problem satisfies this criterion, yet Poincaré proved that three-body systems admit no such global solutions. This has spawned the widespread misconception that "the three-body problem has no solution". This paper proposes a characterization framework for many-body solutions based on Fourier series theory, transforming the trajectories of three-body and N-body systems into superpositions of multiple circles on the complex plane. The Fourier functional forms for two-body, three-body and N-body systems are derived, and three regimes — periodic, quasi-periodic and chaotic — are rigorously distinguished: periodic motion corresponds to discrete Fourier series, quasi-periodic motion to multi-frequency Fourier series, and chaotic motion to Laplace characterization in the complex frequency domain. This paper demonstrates that solutions to the three-body problem are not absent; their representation merely transitions from finite elementary closed expressions to infinite-frequency-domain geometric spectra. The Fourier-series solution preserves the determinism and geometric properties of the Newtonian framework, while echoing Einstein’s pursuit of determinism and geometric unification in physical systems. The Sun–Earth–Moon system is presented as an example to illustrate the geometric configuration of multiple-circle superposition. This study shows that graphical representations themselves constitute a legitimate form of solution. Insisting that the three-body problem must possess point-wise analytic formulas identical to those of the two-body problem violates the diversity of natural laws.

Keywords: three-body problem; many-body dynamics; Fourier series; Laplace transform; 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 formed an inherent aesthetic criterion: a genuine physical solution must be a finite, elementary, closed analytic formula. The two-body problem achieved spectacular success — orbits are conic sections, and position can be directly computed via r = p/(1+e\cos\theta). This standard subsequently became a paradigm.

When researchers attempted to extend this paradigm to the three-body problem, they encountered a fundamental obstacle. Poincaré proved that three-body systems do not possess sufficiently many independent analytic integrals to reduce the system to an integrable form, and hence admit no global elementary closed-form solutions valid for all time. This conclusion is often oversimplified and misinterpreted 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 of expression merely transitions from finite single-circle formulas to infinite multi-circle spectra. This shift does not overturn the deterministic framework of Newtonian mechanics; instead, it aligns at a higher level with Einstein’s ideas regarding the geometrization and determinism of the physical world.

2 Mathematical Foundations: Multi-circle Superposition via Fourier Analysis

2.1 Single Circular Motion

Uniform circular motion on the complex plane is written as:
z(t) = R e^{i\omega t}
where R denotes amplitude and \omega 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 (the ratios of frequencies are not all rational numbers), the motion is quasi-periodic and can be expressed by 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 is the general representation for three-body and many-body systems in the quasi-periodic regime.

2.4 Chaotic Motion and Laplace Transform

When a system exhibits chaotic behavior, trajectories are exponentially sensitive to initial values, and Fourier series fail to converge. The Laplace transform is then introduced:
Z(s) = \int_{0}^{\infty} z(t) e^{-st} dt, \quad s = \sigma + i\omega
Let the maximum Lyapunov exponent be \lambda > 0. The upper bound for growth of z(t) is C e^{\lambda t}. The condition for convergence of the Laplace integral reads:
\int_{0}^{\infty} C e^{\lambda t} e^{-\sigma t} dt = C \int_{0}^{\infty} e^{-(\sigma-\lambda)t} dt < \infty
which holds if and only if \sigma > \lambda. Therefore the boundary of the convergence domain equals the maximum Lyapunov exponent:
\sigma_0 = \lambda
If Z(s) has a pole at s_0 = \lambda + i\omega_0, then z(t) contains a component e^{\lambda t}\cos(\omega_0 t). The real part of the pole encodes the exponential growth rate, and the imaginary part encodes oscillation frequency.

When the convergence domain contains the imaginary axis (\lambda < 0; or \lambda = 0 and the convergence domain strictly contains the imaginary axis), set s = i\omega. The Laplace transform degenerates to the unilateral Fourier transform:
Z(i\omega) = \int_{0}^{\infty} z(t) e^{-i\omega t} dt
Thus the Fourier transform is a special case of the Laplace transform evaluated on the imaginary axis. Periodic and quasi-periodic motion (\lambda \le 0 with the imaginary axis inside the convergence region) are handled by Fourier methods, while chaotic motion (\lambda > 0) is handled by the Laplace transform. Together they form the complete frequency-domain characterization of three-body solutions.

Spectral characteristics of three types of motion:

Type of motion Transform Spectral feature
Periodic Fourier series Discrete spectrum
Quasi-periodic Multi-frequency Fourier series Discrete spectrum
Chaotic Laplace transform Poles in complex frequency domain (real part =  )

3 From Two-body to Three-body: Fourier Characterization

3.1 Two-body System

Take the Sun as reference. The relative motion of Earth’s revolution in the two-body problem yields a Keplerian elliptical orbit. Let the semi-major axis be a, eccentricity e, mean angular velocity \omega_0. Expanding the elliptical orbit on the complex plane into Fourier series:
z_2(t) = \sum_{k=-\infty}^{+\infty} c_k e^{ik\omega_0 t}
where coefficients are
c_k = a \cdot J_k(ke)\ (k \ne 0),\quad c_0 = -\dfrac{a e}{2}
J_k denotes the k-th order Bessel function. As e \to 0 (circular orbit limit), J_k(0)=0 for k\ne\pm1. Only terms k=\pm1 remain, reducing to single-circle superposition: z_2(t) \to a e^{i\omega_0 t}. When e\ne0, Bessel functions of all orders are non-zero, and infinite harmonic components are superimposed to form an elliptical trajectory.

Geometric result: the synthesized trajectory is an ellipse, corresponding to the classical two-body orbit.

3.2 Three-body System

The Sun–Earth–Moon system is a three-body system. Its motion falls into three regimes:

Case 1: Periodic motion. Near special periodic orbits of the three-body problem, motion can be expanded into discrete Fourier series:
z_3(t) = \sum_{k=-\infty}^{+\infty} c_k e^{ik\omega_0 t}

Case 2: Quasi-periodic motion. In regions where KAM tori exist, motion is formed by superposition of multiple incommensurable frequencies:
z_3(t) = \sum_{k_1,k_2,k_3} c_{k_1k_2k_3} e^{i(k_1\omega_1+k_2\omega_2+k_3\omega_3)t}
Geometrically this manifests as multi-layered intertwined orbits.

Case 3: Chaotic motion. For most initial values, the three-body system behaves chaotically. The Laplace transform is used for characterization:
Z_3(s) = \int_{0}^{\infty} z_3(t) e^{-st} dt, \quad \sigma > \lambda
The boundary of the convergence domain equals the maximum Lyapunov exponent, and pole positions encode transient growth rates and oscillation frequencies.

Unified picture of three-body solutions: In periodic and quasi-periodic regimes, three-body motion is expressed as geometric spectra of multi-circle superposition. In chaotic regimes, geometric spectra enter the complex frequency domain and are described by Laplace transforms. The three regimes together constitute the complete frequency-domain description of three-body solutions.

3.3 N-body System

Under quasi-periodic approximation, an N-body system can be written as:
z_N(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}
N=2 recovers the two-body case; N=3 recovers the three-body case. When frequencies are incommensurable, trajectories exhibit quasi-periodic features. When the system enters the chaotic regime, the representation switches to the Laplace complex frequency domain.

4 Transformation to Newtonian Formulation

4.1 What Newton Required

Newton’s criteria for closed-form solutions contain three points:

1. Finite number of terms: no infinite series;
2. Elementary functions: only addition, subtraction, multiplication, division, powers, roots, trigonometric, exponential and logarithmic functions;
3. Closed form: direct evaluation upon substitution of initial conditions.

The two-body system satisfies all three criteria, while the three-body system does not.

4.2 What Fourier Solutions Retain

Although Fourier series contain infinitely many terms, they preserve two core traits of Newtonian solutions:

Newtonian closed-form two-body solution Fourier spectral solution
Finite terms Yes
Elementary functions Yes
Determinism Yes
Geometric nature Yes

Key point: every term R_k e^{i\omega_k t} in the Fourier series is a geometric circle. Given initial conditions, all coefficients R_k,\omega_k are uniquely determined. There is no randomness.

The three-body problem does not lack solutions; its geometric solution is a natural extension of the two-body geometric solution into the infinite-dimensional frequency domain.

4.3 The Accuracy–Globality Dilemma

Newton’s paradigm demands that a solution simultaneously satisfy:

- Accuracy \varepsilon \to 0 (no truncation, infinite terms, exact)
- Globality T \to \infty (valid for all time)
- Finite, elementary, closed form

In practical computation: to achieve accuracy, truncation is required (finite N); to achieve globality, truncation cannot be performed (N\to\infty). Once truncated, globality breaks; with global representation, direct computation becomes infeasible.

For chaotic systems, truncation error grows exponentially with time: \varepsilon(t) \sim \varepsilon_0 e^{\lambda t}. For any finite N, there exists a time T^* such that for t>T^*, error exceeds the specified tolerance. Therefore finite truncation and global validity cannot be simultaneously satisfied.

Newton’s ideal of a "globally exact closed-form solution" is mathematically impossible for chaotic systems.

More precisely, a solution is not an unconditional formula but a quadruple with an accuracy contract:
\text{Solution} = (S,\ \Omega,\ \varepsilon,\ T)
The Newtonian closed-form solution is an ideal limit: S= elementary function, \Omega= global domain, \varepsilon=0, T=\infty. The Fourier–Laplace spectral solution is a finite contract: S=\{c_k,\omega_k\} or Z(s), \Omega= quasi-periodic or finite-time domain, \varepsilon= truncation error, T= controllable.

5 Einstein’s Deterministic Geometry

5.1 God does not play dice

Einstein insisted on the fundamental determinism of the physical world.

The Fourier multi-circle superposition model shows that chaos in three-body orbits does not originate from intrinsic randomness in physics; it arises from the superposition of infinitely many deterministic circular motions. Although the equations are deterministic, chaotic systems are extremely sensitive to initial values, rendering long-term orbit prediction impractical. Given initial conditions, Fourier spectral coefficients are fully determined, and position at any time is strictly determined by the series. This embodies determinism at the fundamental level.

5.2 Geometrization of Physics

Einstein spent his life advancing the geometrization of physics. The complex-plane Fourier expression:
z(t) = \sum_{k} R_k e^{i\omega_k t}
Each term corresponds to a geometric object (circle), and overall motion is the superposition of geometric objects. Three-body motion is geometric weaving in the frequency domain.

5.3 Unification of Fields and Waves

Fourier series form the foundation of field theory and wave analysis. Transforming discrete celestial motion into frequency-domain spectra essentially elevates the many-body problem to a field description. Einstein’s search for a unified field theory finds a unified frequency-domain descriptive approach for many-body motion via Fourier geometric spectra.

6 Graphical Solutions: Why Graphs Qualify as Solutions

6.1 Limitations of the traditional definition of "solution"

Within traditional dynamics, only analytic formulas count as "genuine solutions". Numerical results and geometric images are treated merely as approximations or auxiliary tools.

6.2 Three-fold legitimacy of graphical solutions

Dimension Argument
Mathematics One solution admits two representations: computational form (series, explicit differential equations) and geometric representation (graphs, manifolds, phase-space structures). They are not derivative relations but two representations of the same solution. Graphs exist independently of series and therefore constitute one primitive representation of a solution.
Physics Phase-space geometric solutions (pioneered by Poincaré) are already accepted within academia.
Information Graphs fully carry global configuration, hierarchical order and evolutionary boundaries of the system.

6.3 Example: Sun–Earth–Moon System

System Fourier form Graphical feature
Sun–Earth Superposition of infinite harmonics Regular elliptical orbit
Earth–Moon Multi-frequency superposition Nested secondary trajectory
Sun–Earth–Moon Multi-frequency superposition (quasi-periodic) Multi-layered intertwined network

The synthesized trajectory graph of the three-body system is a complete solution for the system.

Geometric spectral solutions are not only a legitimate form of solution but also serve as retrieval indices for numerical solutions. For any many-body system, its Fourier spectral coefficients and geometric configuration can act as characteristic keys. Given new initial conditions, one extracts geometric spectral features first and retrieves the closest known numerical orbit. Numerical solutions and spectral solutions thus form bidirectional mapping: spectra can be extracted from numerical data, and numerical orbits can be reconstructed from spectra. Graphs are not merely solutions; they serve as entry points to numerical solutions.

At the engineering implementation level, a feature database can be built based on this geometric spectral framework. Spectral coefficients, pole parameters and topological invariants are adopted as primary retrieval keys, bound to corresponding initial conditions, precision and valid time windows to achieve rapid orbit reconstruction. Construction of this database belongs to mature engineering implementation and will not be elaborated in this paper.

6.4 Treatment of Chaotic Cases

For chaotic orbits, Fourier series do not converge and discrete spectra do not exist. The Laplace transform is adopted for characterization: the boundary of the convergence domain equals the maximum Lyapunov exponent; the real part of poles encodes exponential growth rates and the imaginary part oscillation frequencies.

In chaotic regimes, "graphical solutions" degenerate into:

- Poincaré sections
- Power spectral density
- Lyapunov exponent spectra
- Fractal dimensions
- Topological invariants

These statistical and topological invariants constitute generalized geometric spectra for chaotic systems. They are still legitimate forms of solution, though the accuracy contract shifts from point-wise error to statistical error or topological invariance.

6.5 Innovations of this Research

1. Dynamical origin: intrinsic chaos of three-body systems causes tiny deviations to amplify exponentially over time, refusing permanent exact characterization by a fixed finite set of elementary trigonometric functions for all orbital evolution.
2. Standard origin: Newton conflated approximations permitted in practical engineering calculations with the strict ideal criteria for theoretical evaluation. He himself used truncation approximations in astronomical computation, yet required "non-truncated, finite-term, strictly valid for all time" as the rigid standard for perfect solutions.
3. Mathematical origin: Poincaré proved that no global elementary closed-form solutions exist for the three-body problem. Therefore this dual standard is mathematically unachievable within three-body dynamics.

The innovation of this work lies primarily in methodological positioning. Fourier series are pre-existing mathematical tools. The novelty of this paper is elevating multi-circle superposition from a conventional orbit-fitting technique to a geometric spectral characterization framework for many-body dynamics. Under this framework, two-body, three-body and N-body motion can be placed under a unified multi-circle vector picture. Meanwhile, geometric graphical configurations are established as legitimate forms of solution. This re-examines the widespread misinterpretation that "the three-body problem has no solution", and connects Newton’s closed-form paradigm, Poincaré’s topological ideas and Einstein’s program of physical geometrization.

This framework cannot fully meet Newton’s ideal closed-form standard, which is also a contribution of the study. There are three underlying reasons: first, three-body systems possess intrinsic chaotic properties; tiny deviations amplify exponentially over time, so no fixed finite set of elementary trigonometric components can permanently and strictly characterize orbital evolution across all time. Second, Newton conflated practical computation and theoretical evaluation standards: he accepted truncation approximations in astronomical calculations, yet imposed rigid requirements of finite elementary closed expressions valid globally for all time as the criterion for perfect solutions. Third, Poincaré’s results have established that no global elementary closed-form solutions exist for the three-body problem, so this dual requirement is fundamentally unachievable. The geometric-spectral representation in this paper abandons this unachievable formal constraint while preserving the deterministic and geometric core of dynamics, offering a characterization path valid for finite time and controllable precision.

7 Conclusion

1. Solution forms are not unique: two-body systems have closed analytic solutions; three-body systems admit Fourier spectral solutions, Laplace spectral solutions, numerical solutions and topological solutions. Graphs constitute a legitimate form of solution.
2. Fourier multi-circle superposition preserves dynamical determinism: chaos in three-body systems arises from superposition of infinitely many deterministic circular motions, with no probabilistic randomness at the fundamental level.
3. From Newton to Einstein: Newton pursued finite elementary closed forms; Einstein pursued determinism and geometrization of physics. The Fourier–Laplace spectral solution inherits both, upgrading single-circle finite solutions to geometric spectra composed of infinite circles.
4. Accuracy and globality are mutually exclusive: truncation is required for accuracy, while truncation cannot be performed for global validity. Newton’s ideal of a globally exact closed-form solution is mathematically impossible for chaotic systems. A solution should be defined as a quadruple with an accuracy contract: (S,\Omega,\varepsilon,T).
5. Numerical solutions and spectral graphs serve as mutual indices: spectra can be extracted from numerical results, and numerical orbits can be retrieved from spectra. Graphs are not merely solutions, but also entry points to numerical solutions.
6. The three-body problem is not unsolvable. Insisting that the three-body problem must possess point-wise analytic closed formulas analogous to those of the two-body problem is an obsession that violates the diversity of natural laws.

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

References

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[2] Fourier J. Théorie analytique de la chaleur[M]. Paris: Firmin Didot, 1822.
[3] Newton I. Philosophiæ Naturalis Principia Mathematica[M]. London: Royal Society, 1687.
[4] Einstein A. Die Grundlage der allgemeinen Relativitätstheorie[J]. Annalen der Physik, 1916, 49: 769-822.
[5] Laskar J. The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones[J]. Icarus, 1990, 88(2): 266-291.
[6] Arnold V I. Small denominators and problems of stability of motion in classical and celestial mechanics[J]. Russian Mathematical Surveys, 1963, 18(6): 85-191.

Figure captions reserved

Figure 1 The two-body Sun–Earth system: elliptical orbit generated by superposition of infinite harmonic components
Figure 2 Earth–Moon nested two-body system: multi-frequency superposition generating the lunar orbit around Earth
Figure 3 Sun–Earth–Moon three-body system: multi-circle superposition forming multi-layered intertwined quasi-periodic orbit
Figure 4 Pole distribution in the complex Laplace frequency domain for chaotic orbits; real part of poles corresponds to Lyapunov exponent