Three-Body Simulator
2026-10-13·
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7 min read
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阅读量 次
XIETU
m1 Specific Energy
Bound
E₁ = --
½|v|² + U/m
m2 Specific Energy
Bound
E₂ = --
½|v|² + U/m
m3 Specific Energy
Bound
E₃ = --
½|v|² + U/m
1.0×
Center (0.00, 0.00)t = 0.00
0.005
⚙ Initial Conditions
Position, velocity, mass (dimensionless units)
Body 1
m₁ = mass; x₁,y₁ = position; vx₁,vy₁ = velocity
Body 2
m₂ = mass; x₂,y₂ = position; vx₂,vy₂ = velocity
Body 3
m₃ = mass; x₃,y₃ = position; vx₃,vy₃ = velocity
m1
m2
m3
✧ m1×m2
✧ m2×m3
✧ m1×m3
★ Strict Concurrence
⚠ Escape
Computing…
🌌 Three-Body Problem Handbook
From Initial Values to Chaos · From Concurrence to Escape
📌 1. Initial Values
In the 2D three-body problem, each body has 4 degrees of freedom: position (x, y) and velocity (vx, vy). Three bodies give 12 initial values, plus 3 masses = 15 parameters.
1.1 Position (x, y)
Coordinates in the 2D plane. In this simulation, 1 unit ≈ 200 pixels on the canvas.
- Center of mass: weighted average of positions. If
Σ mᵢ·rᵢ = 0, the center of mass is at the origin. - Figure-8 solution: center of mass at origin, so the system as a whole does not translate.
1.2 Velocity (vx, vy)
Velocity components in x and y. Affects trajectory shape and total energy.
- Zero total momentum:
Σ mᵢ·vᵢ = 0makes the center of mass stationary — a necessary condition for the figure-8 solution. - Zero total angular momentum:
Σ mᵢ·(rᵢ × vᵢ) = 0prevents overall rotation.
1.3 Mass (m)
The gravitational "charge" of a body. Larger mass = stronger gravity.
- Equal masses (1:1:1): prerequisite for the figure-8 and Lagrange solutions.
- Extreme mass ratio (e.g., 10:1:1): approximates a "binary + planet" system, more stable.
- Normalize total mass = 3: rescale all masses proportionally so their sum is 3.
💡 Physics of the Figure-8 Solution
Discovered by Moore (1993) and Chenciner-Montgomery (2000). Three equal-mass bodies chase each other along a single figure-8 curve, with period
Discovered by Moore (1993) and Chenciner-Montgomery (2000). Three equal-mass bodies chase each other along a single figure-8 curve, with period
T ≈ 6.325.⚛ 2. Physics
2.1 Specific Energy
Energy per unit mass of each body — the key indicator for escape:
Eᵢ = ½|vᵢ|² + Uᵢ/mᵢ = ½|vᵢ|² − G·Σⱼ≠ᵢ (mⱼ / rᵢⱼ)
Note: the simulation introduces a softening parameter SOFTENING = 0.02. In practice rᵢⱼ = √(Δx² + Δy² + SOFTENING²) to avoid numerical singularities at close encounters.
Note: the simulation introduces a softening parameter SOFTENING = 0.02. In practice rᵢⱼ = √(Δx² + Δy² + SOFTENING²) to avoid numerical singularities at close encounters.
- Eᵢ < 0: Bound. The body lacks enough energy to escape.
- Eᵢ > 0: Escaped. The body's speed exceeds the local escape velocity and it flies to infinity.
- Eᵢ = 0: Critical. The body just reaches infinity with zero speed.
2.2 Total Force
Body i experiences the vector sum of gravitational pulls from the other two:
Fᵢ = Σⱼ≠ᵢ G·mᵢ·mⱼ·(rⱼ − rᵢ) / |rⱼ − rᵢ|³
2.3 Gravity Field and Equipotential Lines
The three-body gravitational potential is:
Φ_total = −G·(m₁/r₁ + m₂/r₂ + m₃/r₃)
Equipotential lines (Gravity Field toggle) are extracted via Marching Squares. Deeper potential wells produce denser, brighter contours.
🪐 3. Known Periodic Solutions
3.1 Lagrange Equilateral Solution (1772)
Three bodies always form an equilateral triangle rotating about their common center of mass.
- Stability: depends on mass ratio. For equal masses it is unstable!
- Routh (1875) showed stability requires
β < √2. For equal masses β = √3 ≈ 1.732 > 1.414, hence unstable.
3.2 Euler Collinear Solution (1767)
Three bodies always remain collinear, rotating about the center of mass.
- Unstable: collinearity is a "needle-tip balance" — any perpendicular offset creates a transverse force that grows.
3.3 Figure-8 Solution (2000)
Three equal-mass bodies chase each other along the same figure-8 curve.
- Unstable, but the instability rate is slower than the previous two (characteristic exponent λ ≈ 0.01~0.05).
✧ 4. Concurrence Theorem for Three Gravity Lines
4.1 Definition
The reverse extension of each body's net force direction is called its "gravity line." Do the three lines meet at a single point?
4.2 Mathematical Result
In general, the three gravity lines do NOT concur.
They do so only when:
They do so only when:
[d₁₂, F₂]·[F₁, F₃] = [d₁₃, F₃]·[F₁, F₂]4.3 When Do They Concur?
- Collinear configuration: the three lines coincide (degenerate concurrence).
- Lagrange equilateral solution: all three lines pass through the center of mass.
- Figure-8 at specific phases: discrete concurrence 2–3 times per period.
⚠ About "Approximate Concurrence"
Because
Because
F₁ + F₂ + F₃ = 0 (net force is zero), the three gravity lines statistically point toward some interior region. The three intersections are usually close together, but strict concurrence occurs only in the cases above.🚀 5. Key Factors for Escape
5.1 Core Mechanism: Energy Redistribution
The fundamental cause of escape in a three-body system is energy redistribution via gravitational interaction:
During a close encounter, one body loses energy and is "slowed," while the other gains energy and is "kicked out."
Total energy is conserved, but individual energies are not — gravitational work transfers energy among bodies.
5.2 Three Escape Mechanisms
| Mechanism | Description | Timescale |
|---|---|---|
| Gravitational Slingshot | A binary transfers energy to a single star, which escapes | Single encounter |
| Chaotic Amplification | Tiny initial differences amplify exponentially, randomly falling into escape regions | Several characteristic times |
| Resonance Pumping | Periodic resonances accumulate energy | Multiple encounters |
| Numerical Error (non-physical) | False escape (caused by accumulated integration error, not real physics) | Depends on precision |
5.3 Ultimate Fate
Three-Body → Binary + Escaped Single Star
For the vast majority of initial conditions, the system eventually disintegrates into a binary plus one escaped star.
5.4 Escape Detection in This Simulation
The specific-energy panel shows Eᵢ in real time:
- Bound: green energy bar, labeled "Bound"
- Escaped: card turns red and blinks; the body turns red with a pulsing halo and "⚠ Escape" text
- Hysteresis: E must remain above threshold for several frames before escape is confirmed, avoiding jitter during close encounters
🌪 6. Chaos and Stability
6.1 What Does "Unstable" Mean?
Stability of a dynamical system is quantified by the Lyapunov exponent λ:
- λ < 0: Stable. Perturbations decay.
- λ = 0: Neutral. Perturbations persist.
- λ > 0: Unstable. Perturbations grow exponentially:
δ(t) ~ δ(0)·e^(λt).
6.2 The Chaotic Nature of the Three-Body Problem
All known periodic solutions of the equal-mass three-body problem are Lyapunov-unstable.
This is the fundamental reason the three-body problem is called "chaotic."
This is the fundamental reason the three-body problem is called "chaotic."
6.3 Stability Comparison
| Solution | Lyapunov exponent | Visual instability time |
|---|---|---|
| Euler collinear | λ ≈ 0.1~0.5 | Disintegrates within a few periods |
| Lagrange equilateral | λ ≈ 0.1~0.3 | Disintegrates within tens of periods |
| Figure-8 | λ ≈ 0.01~0.05 | Disintegrates after hundreds of periods |
🎮 7. Usage Guide
7.1 Canvas Controls
- Drag: pan the view
- Scroll / Pinch: zoom
- Auto Fit: adjusts the view to all three bodies (smooth follow, exits when you interact)
- Reset View: back to initial zoom and center
7.2 Display Toggles
| Toggle | Shows |
|---|---|
| Gravity Field | Potential-well halos + equipotential lines (computed in physical coordinates) |
| Body Trails | Recent trajectories (solid lines, no glow) |
| Body History Trails | Full trajectory history (dashed lines, persistent) |
| Intersection Trails | Recent intersection paths (yellow solid lines) |
| Intersection History | Full intersection history (yellow dashed lines, persistent) |
| Intersection Markers | Intersection dots + strict-concurrence ★ |
| Force Arrows & Gravity Lines | Net force vectors + forward extension dashes for each body |
7.3 Initial Values
- Mass (m₁, m₂, m₃): golden input fields, change the gravitational strength
- Position (x, y): body coordinates
- Velocity (vx, vy): velocity components
- Presets: Figure-8, Lagrange Equilateral, Euler Collinear, Random Perturbation
- Normalize Total Mass: rescale masses so their sum = 3
- Apply & Reset: restart with the new initial conditions
7.4 Observation Tips
Watching an escape:
1. Select the "Lagrange Equilateral" preset
2. Watch the specific-energy panel: all E values start negative
3. At the instability moment, one E jumps positive (hysteresis prevents false positives)
4. That body turns red on the canvas with a pulsing ring
5. Use "Auto Fit" to zoom out and track the escaping body
1. Select the "Lagrange Equilateral" preset
2. Watch the specific-energy panel: all E values start negative
3. At the instability moment, one E jumps positive (hysteresis prevents false positives)
4. That body turns red on the canvas with a pulsing ring
5. Use "Auto Fit" to zoom out and track the escaping body
Experiencing chaos:
1. Click the "Random Perturbation" preset
2. Each click produces slightly different initial values
3. Watch how completely different the trajectories become — that's sensitivity to initial conditions
1. Click the "Random Perturbation" preset
2. Each click produces slightly different initial values
3. Watch how completely different the trajectories become — that's sensitivity to initial conditions
🌌 Three-Body Problem · A Symphony of Chaos and Order
Φ_total = −G·Σ mᵢ/rᵢ · F = m·a · E = ½|v|² + U
Φ_total = −G·Σ mᵢ/rᵢ · F = m·a · E = ½|v|² + U
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