Can three bodies repeat?
Follow a single colour around the figure-eight. All three share the same track, a third of a cycle apart. After about 6.326 time units, each returns to its starting state.
FIELD NOTE 17 / THREE BODIES, ONE LAW
The Three-Body Problem
Three bodies pull on one another. A tiny difference can change the story.
t = 0.00 · model time units
The figure-eight
A · mintB · goldC · violet+ Centre of mass
Positions share one distance scale. Bodies are enlarged point markers. White arrows show velocity; the optional gold arrow shows the selected body’s net gravitational acceleration. Arrows have separate scales and capped lengths.
Three equal masses trace one repeating figure-eight. The faint guide is calculated from the same starting state.
Inertial view: the axes do not turn. Switching frames changes only the view.
Follow a body.
- Mass
- 1.000
- Speed
- 1.272
- Net acceleration
- 0.000
Body C in the original simulation. All quantities use model units.
ORDER AND DISTURBANCE
What has survived the observation?
Figure-eight · a periodic special solution.
Observed 0.00 of 200 model time units.
Original
No escape candidate detected so far.
Disturbed copy
Create a comparison to test a disturbance.
A repeating path and resistance to disturbances are different properties.
No detected escape is a finite observation, not proof of stability. A solver stop is inconclusive beyond the last resolved time. The escape check uses the distant-pair approximation described in the model notes.
SAME LAW · ALMOST THE SAME START
How far do the futures drift?
Create a second copy of the current state, with a velocity change to the body you choose.
- Time since the split
- 0.00
- RMS position difference
- 0.000
- RMS pair-distance difference
- 0.000
PositionsPair distances
Pair distances ignore the overall orientation of each arrangement. A phase shift along an orbit can still affect either trace. Both traces cover the whole observation, with older samples thinned as needed. Each factor of ten takes equal vertical space. Zero has no position on a logarithmic axis. Growth can slow down, reverse or saturate; this trace alone is not a measured Lyapunov exponent.
Energy changes hands.
Original simulationKineticPotentialTotal
Kinetic and potential energy exchange. Their sum stays constant.
Check the calculation
- Largest scaled energy error
- 0
- Largest scaled angular-momentum error
- 0
- Smallest current separation · original
- 1.000
Errors compare each run with its own initial state. A small conservation error is a useful check, but does not guarantee a long chaotic trajectory is exact. Close encounters automatically use smaller time steps. How this is calculated ↓
Running. View controls and comparison settings preserve the original motion.
Make a prediction. Change one thing. See what happens.
Model notes & references ↗FOLLOW THE SCIENCE
A simple law.
An extraordinary range of futures.
The three-body problem asks a precise question: given three masses, their positions and their velocities, where will they go under their mutual gravity? The force law is familiar. The difficulty is that every body moves the other two, continuously changing the forces that determine what happens next.
There is no single convenient formula that gives the general motion the way ellipses describe a bound two-body orbit. Special solutions do exist, and we can calculate trajectories numerically. Some repeat; others are highly sensitive to their starting conditions.
START WITH THE ORIGINAL THREE TRIALS
Let your intuition make a prediction.
How much can 0.001 matter?
Start two copies of a compact trio. Change C’s sideways velocity by just 0.001 in one copy. Watch their first close encounter, then the separation plot. Both obey exactly the same force law.
Who leaves with the energy?
Send a visitor toward an orbiting pair. Watch which bodies become partners and compare kinetic, potential and total energy. Can one move away while the other two bind more tightly?
NOW TRY TO DISTURB THE ORDER
Three ways to stay together.
A pair inside a wider orbit.
Observe the default hierarchical triple for 1,000 time units with a 0.001 kick to C. Then bring the outer starting orbit closer. Does it remain a nested system?
Symmetry is only the beginning.
Start with the linearly stable 100:1:1 triangle. Compare the two distance traces as the perturbed copy drifts in phase. Then draft equal masses and restart to see an unstable symmetric solution.
A dance that turns as it repeats.
Watch one complete loop in the rotating frame, then switch to the inertial view. The shape repeats every 9.176 time units, with a 28.667° rotation each cycle. Increase the disturbance and repeat from the beginning.
These trials use a 0.001 +x velocity change to C. Your Play or Pause choice is preserved. Higher playback speeds help with long observations.
Repeating, stable and long-lived mean different things.
A periodic orbit repeats its state after a period. A relative periodic orbit repeats after a rotation is removed. Quasiperiodic motion combines frequencies that do not share one exact repeat time. Stability asks whether nearby starts stay near the same kind of motion after a small disturbance. A finite calculation showing no escape cannot settle that question for all future time.
The hierarchical example begins with a circular inner pair and a wider circular outer orbit in Jacobi coordinates. These are initial conditions, not two independent Kepler orbits imposed on the motion. The full three-body force perturbs both. The default and its 0.001 C-kick comparison showed no escape through 1,000 time units in our checks. This is an observation, not a theorem.
The triangle has an analytic boundary.
β = 27(m₁m₂ + m₂m₃ + m₃m₁)/(m₁ + m₂ + m₃)²
Circular triangle: β < 1 → linear stability; β > 1 → linear instability
Circular triangle: β < 1 → linear stability; β > 1 → linear instability
For masses proportional to R:1:1, the boundary is R = 25 + 18√2 ≈ 50.456. Our triangle controls keep total mass at 3 and side length at 2, so every mass ratio has the same initial rotation rate, ω = √(3/8). The equal-mass triangle is an exact but linearly unstable solution. Even numerical round-off can eventually disturb its perfect symmetry. The boundary itself is marginal; the strict inequality is the useful linear criterion.
Linear stability concerns infinitesimal perturbations of the circular solution. It is not a guarantee for large kicks, and it is not a blanket nonlinear stability result. Vanderbei: stable and unstable Lagrange configurations ↗
The looping orbit repeats in a turning frame.
The looping dance uses example (a) in Table I of Li & Liao (2020), a linearly stable member of the Broucke–Hadjidemetriou–Hénon satellite family. Here “satellite” names a family of orbit patterns; there are still only three bodies. After T = 9.1758282973, the state returns rotated by θ = 0.500325594634 radians. A view rotating uniformly at θ/T removes that accumulated turn and reveals the repeating loops.
The rotating view changes the coordinates used to draw the results. Gravity is still integrated in the inertial frame. It does not stabilize an unstable orbit. Li & Liao: initial conditions and stability classification ↗
No fixed centre is holding the others.
Each body accelerates toward both companions. The two contributions add as vectors. Turn on the net gravity arrow and follow one body: its acceleration generally points somewhere between the other two, weighted strongly toward a close or massive neighbour.
The centre-of-mass cross stays at rest because all forces are internal. There is no external hand steering the dance, and no force pulling the bodies toward that cross.
Three bodies can share a repeating orbit.
In the equal-mass figure-eight, each body follows the same curve, separated by one third of a period. The bodies pass through the crossing at different times, so they do not collide. Chenciner and Montgomery proved the existence of this periodic solution in 2000.
A tiny perturbation need not produce immediate chaos. Compare the figure-eight with the chaotic trio. A slowly accumulating phase difference is different from rapid sensitive separation.
Two near-identical starts can part ways.
In a sensitive encounter, a small velocity difference changes the timing and distance of the next close pass. That changes the gravitational impulse, which changes the next encounter. The same deterministic law can amplify an initially tiny difference.
Set the tiny change to 0.00001 instead of 0.001, then use Restart this setup for a fair comparison from the same beginning. How long do the futures look alike? The logarithmic graph shows small differences before they are visible on the main view. It is a position-distance diagnostic, not a formal chaos test.
Chaotic motion still conserves energy.
As bodies fall together, potential energy becomes more negative and kinetic energy rises. Close encounters redistribute energy. One body can gain enough to move away, leaving a more tightly bound pair.
The total for the complete isolated system stays constant. A body’s individual energy need not. Angular momentum and total momentum are conserved as well. These constraints narrow the possibilities without specifying one simple orbit.
What does “no general solution” really mean?
It does not mean that gravity stops being predictable, or that every three-body system is chaotic. Exact special solutions exist, as do formal series solutions under appropriate conditions. The difficulty is obtaining a practical general description for arbitrary initial states. Numerical integration gives useful predictions over finite times; in chaotic cases, initial uncertainty and numerical error eventually limit the detailed forecast. Musielak & Quarles: a review of the three-body problem ↗
Three bright dots are not a climate model.
This experiment follows idealized gravitational motion. It does not calculate stellar evolution, planetary habitability or the weather of a planet between three stars. Real triples can also be hierarchical: a close pair with a more distant companion.
The mathematics, assumptions & references
One pairwise force law, applied to all three bodies
a⃗i = G ∑j ≠ i mj (r⃗j − r⃗i) / |r⃗j − r⃗i|³
dr⃗i/dt = v⃗i · dv⃗i/dt = a⃗i
dr⃗i/dt = v⃗i · dv⃗i/dt = a⃗i
Here G = 1. Choose a physical mass scale M and distance scale R, and one time unit becomes √(R³/GM); the velocity unit is √(GM/R). No particular stars, years or kilometres are assumed. The model is planar, with Newtonian point masses, no external forces and no gravitational softening. Disc sizes are exaggerated and are not collision radii.
Conserved quantities
K = ½ ∑ mi|v⃗i|² · U = −∑i<j Gmimj/rij
E = K + U · Lz = ∑ mi(xivy,i − yivx,i)
E = K + U · Lz = ∑ mi(xivy,i − yivx,i)
The energy error shown is |E(t) − E(0)| / [K(0) + |U(0)|]. Angular-momentum error uses a characteristic initial mass–length–speed scale rather than dividing by L(0), which is zero for the figure-eight. When comparing two runs, the larger error is reported, with each run checked against its own beginning.
How the twin experiment works
Compare from here copies the original state at the current time. It adds the selected Δv to the chosen body’s x velocity, then subtracts the resulting centre-of-mass velocity change from all three bodies in the copy. This removes uniform drifting without changing relative velocities. Initial positions stay identical. The original is untouched; the two systems do not interact with each other. Each system is then integrated independently to the same model time.
Position difference: D(t) = √[(|r⃗A − r⃗′A|² + |r⃗B − r⃗′B|² + |r⃗C − r⃗′C|²) / 3]
Corresponding body labels are kept; there is no rotation, rescaling or best-fit alignment of the trajectories. The comparison does change the copy’s energy and generally its angular momentum slightly. A second trace takes the RMS difference of the three pair distances AB, AC and BC; this is insensitive to a rigid rotation of the whole arrangement, but is still sensitive to changes in orbital phase and geometry. Both differences are computed from the physical coordinates, independent of the selected view. The separation plot has a logarithmic vertical axis; exact zero is omitted and values below 10⁻¹² lie below the plotted floor. Starting a fresh comparison clears the displayed plot history and trails, while preserving the original dynamical state.
Numerical method and limits
The solver is adaptive Dormand–Prince 5(4), with relative tolerance 2 × 10⁻¹¹ and absolute tolerance 2 × 10⁻¹³ per state component. Steps are at most 0.04 time units and at most 0.12 of the smaller local crossing or gravitational encounter time. Samples are stored every 0.02 time units. The energy plot keeps the latest 60 time units; trails keep up to 60. The comparison plot retains the whole observation with a maximum of about 4,000 samples, halving its temporal sampling density when needed. This plotting reduction does not change the solver steps. Escape-candidate checks occur at the 0.02-unit observation samples. Actual animation speed can be lower during difficult encounters.
If a pair comes within 10⁻⁴ distance units, the solver cannot resolve a step, or the scaled energy error exceeds 10⁻⁵, the experiment stops at the last shared safe state and reports why. It does not invent a bounce, merger or softened force. Each new observation has a selected duration of 200, 1,000 or 2,000 time units. At its end, Continue observing extends the same physical state and histories. Changing the next-duration control alone leaves the active run unchanged. Close approaches between real extended bodies would require additional physics. Conserving energy well is necessary for a useful calculation but does not certify a chaotic trajectory for arbitrarily long times.
The moving-away message uses an instantaneous approximation: an outward-moving third body must be farther than eight pair separations and four distance units, with positive orbital energy relative to a bound pair treated as one distant mass. It suggests escape; it is not a rigorous asymptotic proof. The displayed bound pair is the one with the most negative instantaneous two-body specific orbital energy.
Starting conditions and references
The figure-eight uses the rounded initial conditions and period printed in Figure 1 of Chenciner & Montgomery’s paper (the first two equal-mass labels are swapped). Rounding means the numerical orbit is an approximation. The chaotic trio, pair-plus-visitor and hierarchical triple are reproducible teaching examples, not reconstructions of observed systems. The hierarchical masses are 1, 1 and 0.5; the inner separation is 1 and the default outer Jacobi distance is 6. The triangle is generated from the circular Lagrange solution. The looping orbit uses the published rounded Table I(a) data, translated to its centre-of-mass frame. Open the starting-state editor to inspect or modify their values.
- Chenciner & Montgomery (2000), A remarkable periodic solution of the three-body problem in the case of equal masses · Paper and initial conditions.
- Musielak & Quarles, The three-body problem · periodic, general and restricted problems, numerical methods and chaos.
- Li & Liao (2020), Satellites of the Broucke–Hadjidemetriou–Hénon family of periodic unequal-mass three-body orbits · Table I(a) and its reported linear stability.
- Empirical Stability Boundary for Hierarchical Triples · why hierarchy, masses and orbital elements all matter.
- Kol (2023), Natural dynamical reduction of the three-body problem · the circular equilateral solution and mass criterion.
- Stone & Leigh (2019), A statistical solution to the chaotic, non-hierarchical three-body problem · chaotic encounters and binary-plus-escaper outcomes.