From Euler’s Seven‑Bridge Problem to Fourier Spectra: Methodological Reconstruction of the Three‑Body Problem

Author: Zhang Suhang

Luoyang School of Mathematics

Abstract

When Euler addressed the Königsberg seven‑bridge problem, he discarded the concrete shapes of physical paths and abstracted the system into a topological network of points and lines, founding graph theory. Drawing on this methodology, this paper re‑examines the three‑body problem. Newton required physical solutions to be closed (finite‑term) and global (valid for infinite time). The two‑body problem satisfies this criterion, yet Poincaré proved that no such solution exists for the three‑body problem — a result later popularly misinterpreted as “the three‑body problem has no solution”. This paper argues that the root of this dilemma lies in an intrinsic paradox within the Newtonian paradigm: finite closed form and global exactness cannot be simultaneously satisfied. Following Euler’s strategy of abstraction, the present work represents three‑body trajectories as superposition of multiple rotating circles on the complex plane: z(t)=\sum R_k e^{i\omega_k t}. The study shows that the two‑body case corresponds to superposition of two circles; the three‑body case requires an infinite set of circular components, with three dominant‑frequency terms available for low‑order approximation; and the N‑body case corresponds to superposition of infinitely many circles. Finite‑order truncation guarantees quantitative accuracy over limited time intervals, while the infinite‑series form secures global determinism. The Fourier‑spectrum solution preserves the determinism and geometric nature of Newtonian mechanics and responds to Einstein’s pursuit of determinism. The three‑body problem is not unsolvable; rather, the form of its solution transitions from “finite closed form” to “infinite spectrum”. Demanding a two‑body‑style single‑point analytic formula for the three‑body system amounts to an obsession confined to a single paradigm.

Keywords: three‑body problem; Fourier series; Euler’s seven‑bridge problem; Newtonian paradigm; diversity of solutions; geometric spectrum

1 Introduction: Two Historical Milestones

Two historically unrelated events share profound methodological parallels.

The first is Euler and the seven‑bridge problem. In eighteenth‑century Königsberg, citizens sought a path that would cross all seven bridges without repetition. Countless attempts based on enumerating concrete routes all failed. Instead of dwelling on bridge lengths, shapes or geographical details, Euler abstracted the system into four vertices (landmasses) and seven edges (bridges). This abstraction exposed the essential parity condition of the problem and gave birth to graph theory and topology.

The second is Newton and the three‑body problem. Using the two‑body problem, Newton established the ideal standard for classical‑mechanical solutions: a closed, finite‑term analytic formula rigorously valid across all time. When later researchers tried to extend this standard to the three‑body problem, they encountered fundamental obstacles. Poincaré demonstrated that no global, elementary, closed analytic solution exists for the three‑body system, a conclusion often simplified into the misleading statement “the three‑body problem has no solution”.

This paper points out that the three‑body dilemma shares the same methodological core as the seven‑bridge problem: when old tools (finite algebraic expressions) reach a dead‑end, one should not persist fruitlessly but instead adopt an alternative descriptive language. Euler replaced physical paths with points and lines; this paper replaces single‑point closed‑form formulas with Fourier multi‑circle spectra.

2 The Intrinsic Paradox of the Newtonian Paradigm

Newtonian mechanics set three criteria for physical solutions:

1. Closed: finite terms, elementary functions, directly evaluable;

2. Global: rigorously valid over the entire time domain;

3. Deterministic: unique outcome given initial conditions.

The two‑body problem perfectly meets all three criteria, with orbits described by regular conic sections:

r = \frac{p}{1+e\cos\theta}

Nevertheless, these criteria contain internal contradictions for nonlinear systems.

- Closed‑form (finite terms) inevitably requires truncation. Truncation introduces error and cannot guarantee global accuracy.

- Global validity (infinite time) demands coverage of an unlimited time domain, which can only be achieved exactly by infinite series.

- Requiring both closed form and global exactness mathematically amounts to demanding finite terms to precisely describe complex evolution over infinite time. For nonlinear dynamical systems, this cannot be realised simultaneously.

The two‑body problem satisfies Newton’s criteria only as a special case whose solution is naturally finite and closed. The three‑body system exposes this contradiction: it is not that no solution exists, but that the “closed‑plus‑global” standard itself fails for nonlinear systems.

3 Methodological Inspiration from Euler’s Seven‑Bridge Problem

The core of Euler’s approach lies in discarding concrete physical shapes and abstracting relational topological structure.

For the seven‑bridge problem, path lengths, bridge widths and land‑area sizes are all discarded; only the connectivity of vertices and edges remains. This abstraction succeeds because it captures the essential structure of the problem.

We face an analogous predicament for the three‑body problem: Newton’s finite closed‑form formulas cannot cover the infinite time domain. Rather than exhausting efforts to search for such a formula, we follow Euler and switch descriptive frameworks.

Euler transformed bridges into edges and landmasses into vertices.

We transform celestial trajectories into superpositions of rotating circles on the complex plane.

4 Fourier Multi‑Circle Superposition: Geometric‑Spectrum Solution for the Three‑Body Problem

4.1 Circular Motion on the Complex Plane

Uniform circular motion on the complex plane reads:

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

where R denotes amplitude, \omega angular frequency, and t time.

4.2 Fourier Series

Any periodic function z(t) can be expanded into a Fourier series:

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

Its geometric meaning: any periodic motion can be represented as superposition of infinitely many uniform circular motions.

4.3 From Two‑Body to N‑Body

- Two‑body system: z_{2}(t) = R_1 e^{i\omega_1 t} + R_2 e^{i\omega_2 t} (superposition of two circles, naturally finite‑term, hence admitting elementary closed form).

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- Three‑body system: exact orbital representation requires infinitely many rotating‑circle components. A low‑order approximation may employ three dominant‑frequency components: z_{3}(t) \approx R_1 e^{i\omega_1 t} + R_2 e^{i\omega_2 t} + R_3 e^{i\omega_3 t}.

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- N‑body system: z_N(t) = \sum_{k=1}^{\infty} R_k e^{i\omega_k t} (superposition of infinitely many circles).

When frequencies \omega_1,\omega_2,\omega_3 are incommensurable, three‑body trajectories exhibit multi‑layer nesting and intricate winding patterns. All evolution still obeys deterministic superposition rules.

4.4 Truncation Accuracy and Global Determinism

Given an accuracy threshold \epsilon>0, a finite truncation order K(\epsilon) exists such that within a finite time interval:

\left| z(t) - \sum_{|k|\le K} c_k\,e^{ik\omega_0 t} \right| < \epsilon

- Local quantitative accuracy is guaranteed by finite‑term truncation (closed‑form approximation).

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- Global determinism is secured by the full infinite‑series representation.

Newton’s demand for both properties simultaneously cannot be satisfied for nonlinear systems. This constitutes the essence of the “closed‑global” paradox.

5 Case Study: Sun‑Earth‑Moon

Treat the Sun, Earth and Moon as sources of dominant spectral components; the composite trajectory yields multi‑layered, intricately winding geometry.

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 Approximation by three dominant‑circle components Multi‑layer intertwined network 

The composite graphical trajectory of the three‑body system constitutes a complete form of solution for the system. The graph is not merely an auxiliary illustration but the geometric realisation of the Fourier series. Given initial conditions, all coefficients R_k,\omega_k are fully determined, with no randomness or probabilistic elements.

6 Dual Responses to Newton and Einstein

6.1 Newton: Preservation of Determinism and Geometric Character

Although the Fourier‑spectrum solution abandons the requirement of finite terms, it retains two core pillars of Newtonian thought:

- Determinism: all coefficients are uniquely fixed by initial conditions.

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- Geometric nature: each term corresponds to a perfect circular motion.

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

6.2 Einstein: “God does not play dice”

Throughout his career Einstein opposed intrinsic randomness in quantum theory and insisted upon a deterministic universe.

Within the Fourier multi‑circle superposition framework: chaos in three‑body orbits originates from nonlinear dynamical coupling; the orbit itself admits representation as superposition of infinitely many deterministic circular motions.

Given initial conditions, all Fourier‑spectrum coefficients are uniquely determined. Positions at arbitrary future instants are strictly prescribed by the series expression. This represents a vindication of determinism.

7 Conclusions: The Diversity of Solution Forms

1. Intrinsic paradox within the Newtonian paradigm: demanding a solution that is both closed (finite‑term) and globally valid across infinite time generates self‑contradiction for nonlinear systems.

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2. Methodological lesson: Euler discarded concrete physical paths and abstracted topology of points and lines; this work abandons single‑point closed‑form expressions and transforms orbital description into frequency‑domain geometric spectra.

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3. The genuine solution form for the three‑body problem is not a finite closed‑form formula but the infinite deterministic multi‑circle spectrum z(t)=\sum R_k e^{i\omega_k t}.

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4. Determinism is preserved: the Fourier spectrum maintains Newtonian determinism and geometric intuition, and responds to Einstein’s advocacy of a deterministic universe.

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5. Graphical representation qualifies as a valid form of solution. Insisting that the three‑body problem must yield two‑body‑style single‑point analytic formulas contradicts the diversity of natural laws.

The order of the universe manifests diverse forms rather than exclusively simple and unique expressions.