What agrees
Energy: −2.25.
Angular momentum: 3.00.
Both orbits satisfy Newton’s equations.
FIELD NOTES / CELESTIAL MECHANICS
Three bodies.
A geometry
of motion.
Gravity connects three paths. In special symmetries, they become a line or a triangle turning as one. Follow the geometry, read its equation, and find the edge of the answer.
Explore the mathematicsWorking paper v0.7 · Website edition 011
Classical exact families and an infinite local series. The general finite closed-form question remains open here.
01 / A SHARED CENTERθ = 15°
The shape of the inquiry
Three ways into
the same question.
What does it mean to solve a motion? Begin with an exact shape, challenge a convincing match, then examine an infinite local series. It does not establish a general finite closed-form solution to the three-body problem.
Read with context. The mathematical derivations are coordinator self-review. The three-model AI study is exploratory and does not establish a reliable model ranking. Read the claims and limitations.
CHAPTER 01 / SHAPE
A shape holds.
A world moves.
These positions come directly from a formula. You are changing its phase and parameters, not watching a numerical integration.
One triangle, three moments.
These are three exact states of the same circular Lagrange orbit: masses (1, 2, 3), side length 1. Every side stays the same length as the triangle turns.
THE LIVE INSTRUMENT
Select a family, then move its phase. The formulas below follow your choice.
The saved figures and explanations are available below. Interactive controls become available when JavaScript finishes loading.
Outer masses = 1. Adjacent gap = 1. G = 1.
Keyboard: arrows move 1°; Shift + arrows move 0.1°; Page Up/Down move 15°; Home/End select 0°/360°.
- Time within this cycle
- 0.3491
- Angular speed
- 1.5000
- Period
- 4.1888
The symmetric Euler family is collision-free for all real time and linearly unstable. An exact orbit need not be stable.
Start with a still view, or take one quarter-turn. Playback targets one cycle per 14 display seconds; a quarter-turn targets 3.5 seconds and pauses. Slow or interrupted rendering can lengthen these durations. Time is measured from phase zero in reference units and returns to zero each cycle. It is not elapsed viewing time. With reduced motion, quarter-turn shows its endpoint immediately.
Motion paused.
The horizontal axis is x; the vertical axis is y; O marks the center of mass. Numbered markers identify bodies and are enlarged on narrow screens; they do not show physical sizes. Exact coordinates remain in the table below.
- Body 1Teal · mass 1
- Body 2Ochre · mass 2
- Body 3Violet · mass 1
Inspect positions and velocities
| Body | Mass | x | y | vₓ | vᵧ |
|---|---|---|---|---|---|
| 1 | 1.0 | −0.86603 | −0.50000 | 0.75000 | −1.29904 |
| 2 | 2.0 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |
| 3 | 1.0 | 0.86603 | 0.50000 | −0.75000 | 1.29904 |
Mathematics, written in motion
A line becomes an orbit.
For the symmetric Euler family, three positions share one rotating direction. The outer bodies mirror each other; the middle body rests at their center.
The minus and plus signs place the outer bodies on opposite sides of the center. The distance a stays fixed.
The rhythm of the rotation
- a
- Gap from the center to either outer body
- m
- Mass of each outer body
- m₀
- Mass of the middle body
- G
- Gravitational constant
Here θ is the phase, ω is the angular speed, and t is physical time. Written in the center-of-mass frame, with positive masses and a > 0. Phase is zero at t = 0; the slider selects a state on that orbit. Matching initial velocities are required. This symmetry describes a special family, not arbitrary initial conditions.
The rhythm of the rotation
- a
- Side length of the equilateral triangle
- M
- Total mass: m₁ + m₂ + m₃
- c
- Center of mass: (m₁s₁ + m₂s₂ + m₃s₃) / M
- R
- Counterclockwise rotation by phase θ
The initial vertices are s₁ = (0, 0), s₂ = (a, 0), s₃ = (a/2, √3 a/2). Subtracting c centers the triangle. R(θ) sends (x, y) to (x cos θ − y sin θ, x sin θ + y cos θ). Here G = 1, a = 1, and masses are (1, 2, 3), so ω = √6. The motion lies in z = 0. Matching initial velocities are required: vᵢ = ω(−yᵢ, xᵢ, 0). This is the circular Lagrange family; arbitrary initial data need not preserve this triangle.
CHAPTER 02 / EVIDENCE
Look beyond
matching numbers.
A shifted circular orbit has the same energy and angular momentum—and still misses the requested initial state. Move the shift to see what invariant checks leave out.
What still needs checking
Do the positions and velocities match the requested state at the requested time?
- Energy, both orbits
- −2.25
- Angular momentum, both
- 3.00
- Maximum position difference / a
- 1.4142
Filled bodies: requested reference.
Outlined bodies: shifted comparison.
Dashed segments: position differences.
Both satisfy Newton’s equations. A check of the intended initial positions and velocities detects the mismatch.
G = 1; masses (1, 2, 1); a = 1. This is a constructed analytical control, not a measured software failure.
Shift 90°. Maximum position difference / a: 1.4142. The initial states differ. Energy and angular momentum are unchanged.
CHAPTER 03 / BOUNDARY
Every guarantee
has a domain.
The paper’s infinite local series covers general distinct-position initial data. Its truncation bound is explicit, but guaranteed only inside a conservative local radius.
READ THE BOUND
Two parts of a guarantee.
At a fixed time fraction, more terms make this upper bound smaller. Near the edge of the guaranteed interval, the bound grows.
bN is the position bound divided by δ.
η = |t| / T < 1 · N is the truncation degree.
Position bound / δ1.953 × 10⁻³
Bound / δ = ηN+1 / [2(1 − η)], where η = |t|/T < 1 and δ is the initial minimum separation divided by 16. Vertical values below 10⁻¹⁵ are clipped on the plot; the readout retains them. This excludes floating-point roundoff and does not represent a measured error or a collision time.
Read sample bounds as a table
| Time fraction η | Position bound / δ |
|---|---|
| 0.25 | 2.543e-06 |
| 0.50 | 1.953e-03 |
| 0.75 | 1.502e-01 |
| 0.95 | 6.302e+00 |
In gratitude
Every equation carries
a human history.
To the observers who kept records, the mathematicians who found patterns, the teachers who made difficult ideas approachable, and the builders who gave others new ways to see.
This work stands within that human history. Some contributors are celebrated; many are unnamed. Their care made these questions ours to inherit. We offer this study with respect for what came before, and excitement for what people may discover next.
An original dedication for this project—not a quotation from a historical source.
A few landmarks in a much wider history
An unfinished conversation.
- 1687
Newton
The Principia joins laws of motion and universal gravitation into a mathematical account of celestial motion.
- 1767
Euler
Collinear configurations reveal exact special solutions: three bodies can remain on a rotating line.
- 1772
Lagrange
Equilateral configurations offer another remarkable family, including unequal masses.
- LATE 1800s
Poincaré
Qualitative dynamics brings new ways to understand complexity beyond a single explicit formula.
- 1912
Sundman
Convergent-series work expands what an exact representation can mean. An infinite series is distinct from a finite formula.
- AN OPEN FUTURE
All of us
Better explanations, careful computation, independent review, and new questions carry the work forward.
Selected landmarks, not a complete history or claim of priority. See the reading notes. The classical families belong to the history of mathematics; this project contributes its documented exploration and reference resources.
Read the research
Follow the idea
to its workings.
What the mathematics establishes
Explicit classical circular families, the fixed-planar-shape boundary, linear instability of the symmetric Euler family, and an infinite local series with a convergence radius and truncation bound.
Download the manuscript · Markdown ↓Download the worked reference card ↓
What the AI study observes
Three original responses under a shared prompt and profile, followed by an unblinded synthesis. A reported intake deviation is retained. The study supports inspection of these outputs, not a model ranking.
The full local record preserves those responses and review history.
What remains open
No general finite closed form, independent mathematical certification, protected examination, numerical trajectory study, or measured learning or deployment benefit is established.
All-time special-family motion is distinct from a general local series. A display check is distinct from a mathematical examination.
About this website edition
Website edition 011 is a visual companion to manuscript v0.7. The included Markdown edition carries its scientific main text, declarations, and reading notes, with internal file links and local operational records removed from the deployable files. The full research record retains the original model responses, configurations, and review history. This page explains the results; the manuscript provides their assumptions and limitations.
Excitement for what comes next
More people able to
see, question & contribute.
The next step can be modest and meaningful: help someone catch a wrong calculation, understand a difficult distinction, or follow an idea all the way back to its evidence.
01
Build trust by checking
Compare an independent implementation with exact reference cases, including examples designed to expose misleading agreement.
02
Make understanding shareable
Pair motion with equations, words, and inspectable states. Then ask whether people learn something they can use elsewhere.
03
Keep the invitation open
Credit inherited knowledge, preserve uncertainty, and give future contributors a clear place to begin.
These are directions for future evaluation. Their educational, accessibility, efficiency, and societal effects have not been measured.
Reading notes & acknowledgment
The ideas have a history.
- Newton, I. Philosophiæ Naturalis Principia Mathematica (1687). Historical landmark.
- Marchal, C. The Three-Body Problem. Elsevier (1990). Background on classical celestial mechanics and solution families.
- Sundman, K. F. “Mémoire sur le problème des trois corps.” Acta Mathematica 36, 105–179 (1912). Historical series representation.
Bibliographic context, not an exhaustive literature or priority review. The cited full texts were not newly retrieved during this website epoch. The local Taylor bound shown here is not a recreation of Sundman’s full theorem.
Ergentics, LLC directs this project. Codex supplied mathematical exposition, coding, and editorial assistance. Classical mathematical contributions are credited to their history; the current derivations and presentation are coordinator self-review.