{ "openapi": "3.1.1", "info": { "title": "N-Body \u0026 Three-Body Orbit Simulator API", "version": "1.0.0", "summary": "Three-body periodic orbits and your own planar n-body systems, with an energy-drift report.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://prooflinenbody.usefulweb.app" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Three-body periodic orbits and your own planar n-body systems, with an energy-drift report.", "description": "Simulate point masses under Newtonian gravity in a plane: published three-body periodic orbits (figure-eight, butterflies, moths, yin-yang and more from Šuvakov \u0026 Dmitrašinović 2013 and the 10-digit refinements of Li \u0026 Liao 2017), Lagrange’s triangle, the chaotic Pythagorean problem, or up to 12 bodies of your own. Choose an adaptive Dormand–Prince RK45 or a symplectic Verlet or Yoshida integrator and get the orbit plot, the energy, momentum and angular-momentum drift, the closest approach and how far a periodic orbit is from its start after each full period. Part of the Proofline math notebook.\n\nMethodology:\n- Each body feels a = Σ G·mⱼ·(rⱼ − rᵢ)/(|rⱼ − rᵢ|² + ε²)^{3/2} from all others (pairwise, O(n²)); positions and velocities form one state vector.\n- RK45 is the numerical engine’s Dormand–Prince 5(4) with adaptive steps (relative tolerance as set, absolute tolerance 100× smaller). Velocity Verlet (kick–drift–kick) and Yoshida’s fourth-order composition of three Verlet steps (weights 1/(2 − 2^{1/3}) and −2^{1/3}/(2 − 2^{1/3})) are symplectic: their energy error stays bounded instead of drifting, as long as the step resolves the closest approaches.\n- Conservation checks: total energy E = Σ ½mv² − Σ G·mᵢmⱼ/r, momentum and angular momentum are compared with their initial values at every output point. For periodic presets the step is aligned with the period and the phase-space distance |state(k·T) − state(0)| is reported for each whole period.\n- Verified: the Chenciner–Montgomery figure-eight returns to within 1e-7 after one period; the 10-digit Li–Liao orbits close to about 1e-8; the 5-digit Šuvakov–Dmitrašinović values close to about 1e-3, as their precision allows; Lagrange’s triangle rotates rigidly; and in the Pythagorean problem the lightest body (m = 3) escapes while the other two form a binary, as Szebehely and Peters found.\n\nA teaching and exploration tool: planar point masses only (no collisions, no relativity, no tides). Chaotic systems amplify every rounding error, so long runs show one possible future, not the future.", "tags": [ "apps" ], "x-mcp-tool": "proofline_nbody", "x-owner-app": "proofline", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineNbodyInput" }, "examples": { "example1": { "summary": "Figure-eight choreography, one period", "value": { "preset": "figure-eight" } }, "example2": { "summary": "Butterfly family, 10-digit initial conditions (closes to 1e-9)", "value": { "preset": "butterfly-refined" } }, "example3": { "summary": "Moth I with the Yoshida symplectic integrator", "value": { "preset": "moth-1", "method": "yoshida4", "step": 0.0002 } }, "example4": { "summary": "Pythagorean problem: chaos, then an escape", "value": { "preset": "pythagorean", "duration": 70 } }, "example5": { "summary": "Your own system: a star with two planets", "value": { "preset": "custom", "bodies": "Star 1 0 0 0 0\nInner 0.001 1 0 0 1\nOuter 0.0005 2.5 0 0 0.632455532", "duration": 50 } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineNbodyResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProoflineNbodyInput": { "type": "object", "additionalProperties": false, "properties": { "preset": { "title": "Initial conditions", "type": "string", "enum": [ "custom", "figure-eight", "butterfly-1", "butterfly-2", "bumblebee", "moth-1", "moth-2", "butterfly-3", "moth-3", "goggles", "butterfly-4", "dragonfly", "yarn", "yin-yang-1a", "yin-yang-1b", "yin-yang-2a", "yin-yang-2b", "figure-eight-refined", "butterfly-refined", "moth-refined", "goggles-refined", "dragonfly-refined", "yin-yang-refined", "lagrange", "pythagorean" ], "default": "figure-eight" }, "bodies": { "title": "Bodies (custom: one per line)", "description": "One body per line: an optional name, then mass, x, y, vx and vy (G = 1 units unless you change G). At most 12 bodies. Used when “My own bodies” is selected.", "type": "string", "maxLength": 4000, "default": "" }, "duration": { "title": "Duration (time units)", "description": "Empty: one period for the periodic presets, 70 for the Pythagorean problem, 10 for your own bodies.", "type": "number", "minimum": 0 }, "method": { "title": "Integrator", "type": "string", "enum": [ "rk45", "yoshida4", "verlet" ], "default": "rk45" }, "tolerance": { "title": "Tolerance (RK45)", "description": "Relative error per step of the adaptive method.", "type": "number", "minimum": 1e-14, "maximum": 0.001, "default": 1e-11 }, "step": { "title": "Step size (fixed-step methods)", "description": "Empty: the period (or the duration) divided by 20,000.", "type": "number", "minimum": 0 }, "G": { "title": "Gravitational constant G", "type": "number", "minimum": 0, "default": 1 }, "softening": { "title": "Softening length", "description": "Plummer softening ε: the force uses r² + ε², which tames close encounters but changes the physics slightly.", "type": "number", "minimum": 0, "default": 0 }, "outputPoints": { "title": "Output points", "type": "integer", "minimum": 11, "maximum": 4001, "default": 801 } }, "required": [] }, "ProoflineNbodyResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "preset", "presetName", "source", "bodies", "G", "softening", "method", "duration", "step", "steps", "forceEvaluations", "rejectedSteps", "completed", "energy0", "energyDriftMax", "energyDriftFinal", "momentumDrift", "angularMomentum0", "angularMomentumDrift", "closestApproach", "period", "returnDistances", "final", "samples" ], "properties": { "preset": { "type": "string" }, "presetName": { "type": [ "string", "null" ] }, "source": { "type": [ "string", "null" ] }, "bodies": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "name", "m", "x", "y", "vx", "vy" ], "properties": { "name": { "type": "string" }, "m": { "type": "number" }, "x": { "type": "number" }, "y": { "type": "number" }, "vx": { "type": "number" }, "vy": { "type": "number" } } } }, "G": { "type": "number" }, "softening": { "type": "number" }, "method": { "type": "string", "enum": [ "yoshida4", "verlet", "rk45" ] }, "duration": { "type": "number" }, "step": { "type": [ "number", "null" ] }, "steps": { "type": "integer" }, "forceEvaluations": { "type": "integer" }, "rejectedSteps": { "type": "integer" }, "completed": { "type": "boolean" }, "energy0": { "type": "number" }, "energyDriftMax": { "type": "number" }, "energyDriftFinal": { "type": "number" }, "momentumDrift": { "type": "number" }, "angularMomentum0": { "type": "number" }, "angularMomentumDrift": { "type": "number" }, "closestApproach": { "type": "number" }, "period": { "type": [ "number", "null" ] }, "returnDistances": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "periods", "time", "distance" ], "properties": { "periods": { "type": "integer" }, "time": { "type": "number" }, "distance": { "type": "number" } } } }, "final": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "name", "x", "y", "vx", "vy" ], "properties": { "name": { "type": "string" }, "x": { "type": "number" }, "y": { "type": "number" }, "vx": { "type": "number" }, "vy": { "type": "number" } } } }, "samples": { "type": "object", "additionalProperties": false, "required": [ "t", "xy", "energyError" ], "properties": { "t": { "type": "array", "items": { "type": "number" } }, "xy": { "type": "array", "items": { "type": "array", "items": { "type": [ "number", "null" ] } } }, "energyError": { "type": "array", "items": { "type": "number" } } } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }