Gravity · initial conditions

Three-Body Gravity

Explore the figure-eight choreography, a rotating Lagrange triangle, or a visitor passing a binary. Nudge one body, replay both trajectories together, and check the numerical record against 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. TWO INITIAL STATES.

Change the beginning

Recorded pathsColours identify the three bodies throughout replay.Unperturbed referenceDashed paths use the same masses with zero 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

Normalized units with gravitational constant one. Unsoftened Newtonian point masses in a plane, initially at rest in the barycentric frame. The published figure-eight requires three equal masses and zero perturbation. Both records end if either run reaches pair separation 0.04. A finite-time separation curve alone does not establish chaos.

Reference and perturbed paths are integrated separately with matching masses and times. Separation alone is not proof of chaos, and close encounters never silently reset the system.

Recorded time

t=0.00

The solid record includes the chosen velocity nudge. The dashed reference uses zero nudge and identical masses. Both start in their own stationary barycentric frame. The separation compares matching body identities 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: every body continually changes where the others will feel 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}

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

01
Observe a special periodic solution
Choose Figure-eight and enable trails.
What to observe: Three equal masses follow one closed curve in sequence.
02
Add a perturbation
Choose Nudge the triangle, change the velocity nudge and launch again; compare solid and dashed paths.
What to observe: Separation varies with time. Finite-time growth alone does not prove chaos; perturbations near a stable orbit can stay bounded.
03
Check global constraints
Inspect the white barycentre cross and the energy, momentum and angular-momentum records.
What to observe: Motion is complex while global invariants stay nearly stable.

Three-Body Gravity

How far do the two beginnings separate?

Three bodies: why simple laws do not produce simple orbits