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Sandbox Physics

Gravity · chaos

Three-Body Problem

Three bodies are integrated under mutual gravity. Unlike the two-body case, there is no universal closed-form solution. Start from the famous figure-eight choreography, or nudge it and watch the motion fall into chaos.

Interactive modelThree-Body Problem
Time0.0 s
Total energy+0.000
Total momentum0.000
StatusBounded

Physics tutorial

Three bodies: why simple laws do not produce simple orbits

BackgroundEach of three bodies continuously changes the motion of the other two. The two-body problem reduces to one relative coordinate; a general three-body system does not fully decouple.

Why it mattersIt explains why long-term solar-system stability, stellar systems, and space missions rely on numerical integration rather than one closed formula.

Start with the essentials

Focus question
Newtonian gravity is simple, so why is there no general analytic formula for three-body paths?
One-sentence intuition
The hard part is feedback: every body continually changes where the others will feel future forces.

Core mathematical model

Many-body acceleration

r¨i=G∑j≠imjrj−ri∣rj−ri∣3\ddot{\mathbf r}_i=G\sum_{j\ne i}m_j\frac{\mathbf r_j-\mathbf r_i}{|\mathbf r_j-\mathbf r_i|^3}

Every step recomputes all pairwise forces using the new positions.

Conserved total momentum

P=∑imivi=constant\mathbf P=\sum_i m_i\mathbf v_i=\text{constant}

Internal gravity cancels in pairs, so an isolated center of mass moves uniformly.

Common difficulties

No general closed form does not mean uncomputable

Typical misconceptionWithout a formula, three-body motion cannot be predicted.

Better mental modelNumerical integration predicts finite intervals accurately; what is missing is one simple expression for arbitrary initial states.

Chaos does not break conservation

Typical misconceptionIrregular paths imply energy or momentum is not conserved.

Better mental modelChaotic paths remain constrained by invariants; conservation does not imply periodicity.

Run the experiment

  1. 01

    Observe a special periodic solution

    Choose Figure-eight and enable trails.

    What to observe: Three equal masses follow one closed curve in sequence.
  2. 02

    Add a perturbation

    Increase perturbation from a very small value.

    What to observe: Paths agree briefly, then encounter order and shape separate.
  3. 03

    Check global constraints

    Show the center of mass and inspect energy and momentum.

    What to observe: Motion is complex while global invariants stay nearly stable.