Gravity · initial conditions

Three-Body Gravity

Why does one extra body remove the general orbit formula? Each body's motion keeps changing the forces on the other two, and those forces in turn change its own motion. Apart from a few special solutions, no general closed-form answer exists. The motion has to be integrated numerically and checked against the conserved quantities.

Watch three bodies chase each other around a figure eight, turn as a rotating Lagrange triangle, or meet a visitor passing a binary. Give one body a small push, replay the original and nudged runs together, and check the computed motion against the conserved quantities.

Total energy $0$

Total momentum magnitude $0$

Angular momentum $0$

Closest pair $0$

RMS distance from reference $0$

Energy error / initial potential scale $0$

Three bodies from two starting states

Change how the three bodies start

Recorded pathsEach body keeps its own colour throughout the replay.Unperturbed referenceDashed paths use the same masses with no velocity change.Velocity arrowsArrow lengths are compressed. The white cross marks the barycentre.Third mass

m_3

1

Third-body velocity change · percent

\delta

0

Recorded time span

T

12

Units are normalized so the gravitational constant is one. The bodies are Newtonian point masses in a plane, with no softening, and their barycentre starts at rest. The published figure-eight orbit needs three equal masses and zero perturbation. If any two bodies come within 0.04 of each other in either run, both records end at the same time. A separation curve over a finite time does not by itself establish chaos.

The original and nudged runs are integrated separately, with the same masses and the same times. Paths that drift apart do not by themselves prove chaos. A close encounter never resets the system without telling you.

Recorded time

t=0.00

The solid path includes the velocity nudge you chose. The dashed reference has no nudge and the same masses. Each run starts at rest in its own barycentric frame. The separation compares each body with its counterpart at the same recorded time.

D(t)=\sqrt{\frac{1}{3}\sum_{i=1}^{3}\left|\mathbf r_i(t)-\mathbf r_{i,0}(t)\right|^2}

Time, distance and mass use normalized units. The plane and rings are coordinate guides. Display size does not represent a physical radius. Energy errors are divided by the magnitude of the initial potential energy; they remain meaningful at zero total energy. Replay never recomputes the trajectory.

Return to Kepler orbits →The figure-eight paper MIT: Lagrange solutions

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: each body keeps moving the others, which changes where they will feel their future forces.

Core mathematical model

Many-body acceleration

\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

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

The internal gravitational forces cancel in pairs, so the center of mass of an isolated system moves at constant velocity.

Common difficulties

No general formula does not mean no prediction

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

Better mental modelNumerical integration predicts the motion accurately over a finite time. What is missing is one simple expression that works for every starting state.

Chaos does not break conservation

Typical misconceptionIf the paths look irregular, energy or momentum must not be conserved.

Better mental modelChaotic paths are still limited by the conserved quantities, and conservation does not make the motion periodic.

Run the experiment

01
Observe a special periodic solution
Choose Figure-eight and enable trails.
What to observe: Three equal masses chase one another around the same closed curve.
02
Add a perturbation
Choose Nudge the triangle, change the velocity nudge and launch again; compare solid and dashed paths.
What to observe: The gap between the two paths changes with time. Growth over a finite time does not prove chaos on its own; near a stable orbit, a small perturbation can stay bounded.
03
Check the conserved quantities
Inspect the white barycentre cross and the energy, momentum and angular-momentum records.
What to observe: The motion is complicated, yet the conserved quantities of the whole system stay nearly constant.

Three-Body Gravity

How far apart do the two runs drift?

Three bodies: why simple laws do not produce simple orbits