43:["$","title",null,{"children":"Gravitation and the n-body problem | Mark McClure | Observable"}] 4b:T44c,// Note the value here refers to one of the properties of // the configurations variable. viewof configuration = select({ title: "Configuration", options: [ { label: "Two equal masses", value: "twoEqual" }, { label: "Two unequal masses", value: "twoUnequal" }, { label: "Symmetric, Eulerian central configuration", value: "eulerianSymmetricCentralConfiguration" }, { label: "Asymmetric, Eulerian central configuration", value: "eulerianAsymmetricCentralConfiguration" }, { label: "Sun/Planet/Moon", value: "sunPlanetMoon" }, { label: "Lagrange points", value: "lagrange" }, { label: "Perturbed Lagrange points", value: "perturbedLagrange" }, { label: "Figure 8", value: "figure8" }, { label: "Three Body 0", value: "threeBody0" }, { label: "Three Body 11", value: "threeBody11" }, { label: "Three Body 18", value: "threeBody18" }, { label: "3-4-5 Triangle", value: "threeFourFive" }, { label: "Semi-random", value: "semiRandom" }, { label: "7 stable objects", value: "circle7" } ], value: "twoUnequal" })4c:Te12,orbit = { let nodes = configurations[configuration]; // Might want to set the configuration to something // not yet available in menu using the following line: // nodes = configurations.experimental; let img_width = width < 900 ? width : 900; let aspect = (nodes.ymax - nodes.ymin) / (nodes.xmax - nodes.xmin); let height = img_width * aspect; let div = d3 .create('div') .style('width', `${img_width}px`) .style('height', `${height + 80}px`); let running = true; let interval_id; let controls = div .append('div') .style('width', `${100}px`) .style('height', `${30}px`); let stop_button = controls .append('button') .on('click', function() { if (running) { running = false; clearInterval(interval_id); d3.select(this).html('Start'); } else { running = true; start(); d3.select(this).html('Stop'); } }) .html('Stop'); let container = div .append('div') .attr('id', 'orbit_container') .style('position', 'relative') .style('width', `${img_width}px`) .style('height', `${height}px`); let canvas = container .append('canvas') .style('position', 'absolute') .style('top', 0) .style('left', 0) .attr('width', img_width) .attr('height', height); let ctx = canvas.node().getContext('2d'); ctx.globalAlpha = 0.9; let svg = container .append('svg') .style('position', 'absolute') .style('top', 0) .style('left', 0) .attr('width', img_width) .attr('height', height); let pad = 20; let x_scale = d3 .scaleLinear() .domain([nodes.xmin, nodes.xmax]) .range([pad, img_width - pad]); let y_scale = d3 .scaleLinear() .domain([nodes.ymin, nodes.ymax]) .range([height - pad, pad]); let pts_to_path = d3 .line() .x(d => d.x) .y(d => d.y); let n = nodes.length; let x0 = nodes.map(o => o.x); let y0 = nodes.map(o => o.y); let vx0 = nodes.map(o => o.vx); let vy0 = nodes.map(o => o.vy); let init = x0 .concat(y0) .concat(vx0) .concat(vy0); let sol = [init]; let cnt = 0; function start() { interval_id = setInterval(function() { cnt = cnt + 1; if (cnt > nodes.period) { sol = [init]; cnt = 0; } else { sol = rk4_multi_step( F(nodes), sol[sol.length - 1], 0, nodes.dt, nodes.steps_per_frame ); } let paths = new Array(n); for (let i = 0; i < n; i++) { paths[i] = sol.map(o => ({ x: x_scale(o[i]), y: y_scale(o[i + n]) })); } //svg.selectAll('.temp').remove(); paths.forEach(function(p, i) { ctx.beginPath(); ctx.moveTo(p[0].x, p[0].y); ctx.strokeStyle = d3.schemeCategory10[i]; // 'rgba(0,0,0,0.5)'; ctx.lineWidth = 1; for (let j = 0; j <= nodes.steps_per_frame; j++) { ctx.lineTo(p[j].x, p[j].y); } ctx.stroke(); ctx.closePath(); svg.select(`#point${i}`).remove(); svg .append("circle") .attr('id', `point${i}`) .attr("r", 5) .attr("cx", p[p.length - 1].x) .attr("cy", p[p.length - 1].y) .style("fill", d3.schemeCategory10[i]) .style('stroke', 'black') .style('stroke-width', '1px'); }); if (cnt > nodes.cnt_max) { clearInterval(interval_id); } }, nodes.delay); } start(); invalidation.then(() => clearInterval(interval_id)); return div.node(); }4d:T219e,configurations = { let configurations = {}; // Experimental - you gotta uncomment the line redefining `nodes` // near the top of the definition of `orbit` to access this. // configurations.experimental = [ // { m: 4, x: 0, y: 0, vx: 0, vy: -1 }, // { m: 2, x: 1, y: 0, vx: 0, vy: 0.8824172416557946 }, // { m: 1, x: 1.7186229732164608, y: 0, vx: 0, vy: 2.235165516688411 } // ]; // configurations.experimental.xmin = -.5; // configurations.experimental.xmax = 2; // configurations.experimental.ymin = -1; // configurations.experimental.ymax = 1; // configurations.experimental.steps_per_frame = 4; // configurations.experimental.dt = 0.001; // configurations.experimental.delay = 1; // Two bodies with equal masses configurations.twoEqual = [ { m: 4, x: 0, y: 0, vx: 0, vy: -1 / Math.sqrt(2) }, { m: 4, x: 1, y: 0, vx: 0, vy: 1 / Math.sqrt(2) } ]; configurations.twoEqual.xmin = -0.2; configurations.twoEqual.xmax = 1.2; configurations.twoEqual.ymin = -1 / 2; configurations.twoEqual.ymax = 1 / 2; configurations.twoEqual.steps_per_frame = 1; configurations.twoEqual.dt = 0.001; configurations.twoEqual.delay = 0; // Two bodies with unequal masses configurations.twoUnequal = [ { m: 1, x: 1, y: 0, vx: 0, vy: 2 }, { m: 20, x: 0, y: 0, vx: 0, vy: -0.1 } ]; configurations.twoUnequal.xmin = -0.2; configurations.twoUnequal.xmax = 1.2; configurations.twoUnequal.ymin = -0.4; configurations.twoUnequal.ymax = 0.4; configurations.twoUnequal.steps_per_frame = 1; configurations.twoUnequal.dt = 0.001; configurations.twoUnequal.delay = 0; // A symmetric, Eulerian central configuration // All three points stay on a line segment with one point // at the center (of mass) and the others revolving around it. configurations.eulerianSymmetricCentralConfiguration = [ { m: 4, x: 0, y: 0, vx: 0, vy: -1 }, { m: 0.5, x: 0.5, y: 0, vx: 0, vy: 0 }, { m: 4, x: 1, y: 0, vx: 0, vy: 1 } ]; configurations.eulerianSymmetricCentralConfiguration.xmin = -0.2; configurations.eulerianSymmetricCentralConfiguration.xmax = 1.2; configurations.eulerianSymmetricCentralConfiguration.ymin = -1 / 2; configurations.eulerianSymmetricCentralConfiguration.ymax = 1 / 2; configurations.eulerianSymmetricCentralConfiguration.steps_per_frame = 1; configurations.eulerianSymmetricCentralConfiguration.dt = 0.001; configurations.eulerianSymmetricCentralConfiguration.delay = 0; configurations.eulerianSymmetricCentralConfiguration.period = 844; // An Eulerian asymmetric central configuration // All three points stay on a line segment but // the middle point is not at the center of mass. configurations.eulerianAsymmetricCentralConfiguration = [ { m: 100, x: 0, y: 0, vx: 0, vy: -0.618 }, { m: 10, x: 1, y: 0, vx: 0, vy: 5.419013137068794 }, { m: 1, x: 1.3629038801971198, y: 0, vx: 0, vy: 7.609868629312047 } ]; configurations.eulerianAsymmetricCentralConfiguration.xmin = -1; configurations.eulerianAsymmetricCentralConfiguration.xmax = 2; configurations.eulerianAsymmetricCentralConfiguration.ymin = -1.3; configurations.eulerianAsymmetricCentralConfiguration.ymax = 1.3; configurations.eulerianAsymmetricCentralConfiguration.steps_per_frame = 1; configurations.eulerianAsymmetricCentralConfiguration.dt = 0.001; configurations.eulerianAsymmetricCentralConfiguration.delay = 5; configurations.eulerianAsymmetricCentralConfiguration.period = 293; configurations.threeUnequal = (function() { let nodes = [ { m: 20, x: 0, y: 0, vx: 0, vy: -0.1 }, { m: 1, x: 1, y: 0, vx: 0, vy: 2 }, { m: 0.0001, x: 1.5, y: 0, vx: 0, vy: 3.5 } ]; nodes.xmin = -1; nodes.xmax = 2; nodes.ymin = -1.4; nodes.ymax = 1.4; nodes.steps_per_frame = 5; nodes.dt = 0.001; nodes.delay = 25; return nodes; })(); // Lagrange configurations.lagrange = (function() { let nodes = [ { m: 4, x: 0, y: 0, vx: 0, vy: -1 / Math.sqrt(2) }, { m: 4, x: 1, y: 0, vx: 0, vy: 1 / Math.sqrt(2) }, { m: 0.0000001, x: 1 / 2, y: Math.sqrt(3) / 2, vx: -Math.sqrt(3 / 2), vy: 0 } ]; nodes.xmin = -0.2; nodes.xmax = 1.2; nodes.ymin = -0.2; nodes.ymax = 1; nodes.steps_per_frame = 20; nodes.dt = 0.0001; nodes.delay = 5; nodes.period = 4 * 480; return nodes; })(); // Perturbed Lagrange configurations.perturbedLagrange = (function() { let nodes = [ { m: 4, x: 0, y: 0, vx: 0, vy: -1 / Math.sqrt(2) }, { m: 4, x: 1, y: 0, vx: 0, vy: 1 / Math.sqrt(2) }, { m: 0.0000001, x: 1 / 2 + 0.001, y: Math.sqrt(3) / 2, vx: -Math.sqrt(3 / 2), vy: 0 } ]; nodes.xmin = -0.2; nodes.xmax = 1.2; nodes.ymin = -0.2; nodes.ymax = 1; nodes.steps_per_frame = 20; nodes.dt = 0.0001; nodes.delay = 5; nodes.period = 4 * 480; return nodes; })(); // Figure 8 configurations.figure8 = (function() { let nodes = [ { m: 1, x: -0.97000436, y: 0.24308753, vx: 0.4662036850, vy: 0.4323657300 }, { m: 1, x: 0, y: 0, vx: -0.93240737, vy: -0.86473146 }, { m: 1, x: 0.97000436, y: -0.24308753, vx: 0.4662036850, vy: 0.4323657300 } ]; nodes.xmin = -1.2; nodes.xmax = 1.2; nodes.ymin = -0.6; nodes.ymax = 0.6; nodes.steps_per_frame = 1; nodes.track_length = 150; nodes.dt = 0.01; nodes.delay = 0; return nodes; })(); // Sun, Planet, Moon (seems stable) configurations.sunPlanetMoon = [ { m: 4, x: 0, y: 0, vx: 0, vy: -0.5 }, { m: 0.00001, x: 4 / 5, y: 0, vx: 0, vy: 0 }, { m: 1, x: 1, y: 0, vx: 0, vy: 2 } ]; configurations.sunPlanetMoon.xmin = -1.4; configurations.sunPlanetMoon.xmax = 1.4; configurations.sunPlanetMoon.ymin = -1.4; configurations.sunPlanetMoon.ymax = 1.4; configurations.sunPlanetMoon.steps_per_frame = 5; configurations.sunPlanetMoon.dt = 0.001; configurations.sunPlanetMoon.delay = 5; // http://three-body.ipb.ac.rs/bsol.php?id=0 configurations.threeBody0 = (function() { let nodes = [ { m: 1, x: -0.9892620043, y: 0, vx: 0, vy: 1.9169244185 }, { m: 1, x: 2.2096177241, y: 0, vx: 0, vy: 0.1910268738 }, { m: 1, x: -1.2203557197, y: 0, vx: 0, vy: -2.1079512924 } ]; nodes.xmin = -2.3; nodes.xmax = 2.3; nodes.ymin = -1; nodes.ymax = 1; nodes.steps_per_frame = 1; nodes.track_length = 150; nodes.dt = 0.01; nodes.delay = 0; return nodes; })(); // http://three-body.ipb.ac.rs/bsol.php?id=11 configurations.threeBody11 = (function() { let nodes = [ { m: 1, x: -0.337076702, y: 0, vx: 0, vy: 0.9174260238 }, { m: 1, x: 2.1164029743, y: 0, vx: 0, vy: -0.0922665014 }, { m: 1, x: -1.7793262723, y: 0, vx: 0, vy: -0.8251595224 } ]; nodes.xmin = -2.3; nodes.xmax = 2.3; nodes.ymin = -2.3; nodes.ymax = 2.3; nodes.steps_per_frame = 3; nodes.track_length = 150; nodes.dt = 0.01; nodes.delay = 0; return nodes; })(); // http://three-body.ipb.ac.rs/bsol.php?id=18 configurations.threeBody18 = (function() { let nodes = [ { m: 1, x: 0.8920281421, y: 0, vx: 0, vy: 0.9957939373 }, { m: 1, x: -0.6628498947, y: 0, vx: 0, vy: -1.6191613336 }, { m: 1, x: -0.2291782474, y: 0, vx: 0, vy: 0.6233673964 } ]; nodes.xmin = -1.2; nodes.xmax = 1.2; nodes.ymin = -1; nodes.ymax = 1; nodes.steps_per_frame = 1; nodes.track_length = 150; nodes.dt = 0.01; nodes.delay = 0; return nodes; })(); // 3-4-5 triangle configurations.threeFourFive = (function() { let nodes = [ { m: 3, x: 1, y: 3, vx: 0, vy: 0 }, { m: 4, x: -2, y: -1, vx: 0, vy: 0 }, { m: 5, x: 1, y: -1, vx: 0, vy: 0 } ]; nodes.xmin = -6; nodes.xmax = 6; nodes.ymin = -6; nodes.ymax = 6; nodes.aspect = (nodes.ymax - nodes.ymin) / (nodes.xmax - nodes.xmin); nodes.steps_per_frame = 1000; nodes.dt = 0.00005; nodes.delay = 1; return nodes; })(); configurations.circle7 = circular_configuration(7); configurations.semiRandom = (function() { let n1 = 3; let s1 = 0.2083785248428932; let n2 = 4; let s2 = 0.3889050449181679; let n3 = 3; let s3 = 0.9968774001773553; let n4 = 4; let s4 = 0.2990274913092489; let nodes = random_configuration(n4, s4); return nodes; })(); return configurations; }4e:T917,commentary = ({ twoEqual: "Two equal masses in mutal orbit", twoUnequal: "A basic star/planet model", eulerianSymmetricCentralConfiguration: 'A central configuration is a special arrangment that remains geometrical similar to it\'s initial shape. In the simplest instance, two large masses revolve around an object at the center of mass. The three masses lie on a line segment throughout the evolution.', eulerianAsymmetricCentralConfiguration: 'In this central configuration, the three masses are 100, 10, and 1. They revolve around the center of mass. ', sunPlanetMoon: "A moon in orbit around a large planet in orbit around a sun.", lagrange: 'Lagrange found another type of central configuration whose objects lie at the vertices of an equilateral triangle. In the example here, there are two large bodies are mutual orbit. The central configuration is completed by very small third mass.', perturbedLagrange: 'While stable, the basin of attraction of a Lagrange point can be quite small. Thus, we can move the small point a small amount and lose stability.', figure8: 'There are a surprising number of surprising solutions to the three body problem. I learned of this one from The Three Body Gallery.', threeBody0: 'Another from The Three Body Gallery', threeBody11: 'Another from The Three Body Gallery', threeBody18: 'Another from The Three Body Gallery', threeFourFive: 'When three equal masses are placed at rest the vertices of a 3-4-5 triangle, things get a little crazy', semiRandom: "Generally, a few objects placed at rest don't play nice.", circle7: 'If we place seven objects of equal mass placed evenly around a circle with a little velocity tangential to the circle, the result can be stable.' })4f:T4a5,function F(nodes) { let G = 1; let n = nodes.length; function x_accel(s, i, j) { let m2 = nodes[j].m; let x1 = s[i]; let y1 = s[i + n]; let x2 = s[j]; let y2 = s[j + n]; return (m2 * (x2 - x1)) / ((x2 - x1) ** 2 + (y2 - y1) ** 2) ** (3 / 2); } function y_accel(s, i, j) { let m2 = nodes[j].m; let x1 = s[i]; let y1 = s[i + n]; let x2 = s[j]; let y2 = s[j + n]; return (m2 * (y2 - y1)) / ((x2 - x1) ** 2 + (y2 - y1) ** 2) ** (3 / 2); } function xpp(s, i) { return ( G * d3.sum( d3.range(n).map(function(j) { if (j != i) { return x_accel(s, i, j); } else { return 0; } }) ) ); } function ypp(s, i) { return ( G * d3.sum( d3.range(n).map(function(j) { if (j != i) { return y_accel(s, i, j); } else { return 0; } }) ) ); } function F(s, t) { return s .slice(2 * n, 3 * n) .concat(s.slice(3 * n, 4 * n)) .concat(d3.range(n).map(i => xpp(s, i))) .concat(d3.range(n).map(i => ypp(s, i))); } return F; }50:T408,function circular_configuration(k) { let nodes = d3.range(k).map(i => ({ m: 1, x: Math.cos((2 * Math.PI * i) / k), y: Math.sin((2 * Math.PI * i) / k), vx: -Math.sin((2 * Math.PI * i) / k) / 3, vy: Math.cos((2 * Math.PI * i) / k) / 3 })); nodes.xmin = -2; nodes.xmax = 2; nodes.ymin = -2; nodes.ymax = 2; nodes.steps_per_frame = 50; nodes.dt = 0.001; nodes.delay = 5; if (k == 3) { nodes.period = 169; nodes.delay = 10; nodes.dt = 0.001; nodes.steps_per_frame = 20; } else if (k == 4) { nodes.period = 124; nodes.delay = 10; nodes.dt = 0.001; nodes.steps_per_frame = 20; } else if (k == 5) { nodes.period = 100; nodes.delay = 15; nodes.dt = 0.001; nodes.steps_per_frame = 20; } else if (k == 6) { nodes.period = 86; nodes.delay = 15; nodes.dt = 0.001; nodes.steps_per_frame = 20; } else if (k == 7) { nodes.period = 610; nodes.delay = 2; nodes.dt = 0.0005; nodes.steps_per_frame = 5; } return nodes; }51:[] 47:{"id":"82ce6858b6472d7f","slug":"gravitation-and-the-n-body-problem","title":"Gravitation and the n-body problem","thumbnail":"04f076cebe9a508f793e754d6667e7f051674e8bc3c34d7a7808cc09af3d1818","access_level":"public","create_time":"$D2020-09-11T23:08:02.569Z","update_time":"$D2024-01-05T16:06:15.814Z","view_time":null,"first_public_time":"$D2020-11-09T18:15:52.261Z","has_unpublished_edits":null,"fork":true,"stars":7,"starred":false,"owner":{"type":"team","id":"d771c8ae812d5130","login":"mcmcclur","name":"Mark McClure","avatar_url":"https://avatars-next.observableusercontent.com/e615e819ce83d43faef607e48c53c5ff63943aa0ba9f5848499bd02fef7396f4","tier":"starter_2024","bio":"I'm a professor of mathematics at the University of North Carolina Asheville. 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We use this to present a few examples.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1263,"value":"$4b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1378,"value":"info = html`${commentary[configuration]}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":477,"value":"$4c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1466,"value":"md`## Configurations\n\n\\`configurations\\` is an object and each property of \\`configurations\\` is a an array of objects with mass \\`m\\`, initial position (\\`x\\` and \\`y\\`), and initial velocity (\\`vx\\` and \\`vy\\`) properties. In addition, you can set \\`xmin\\`, \\`xmax\\`, \\`ymin\\`, and \\`ymax\\` properties to define the drawing window. You can also set several other properties of a configuration:\n\n * \\`dt\\`: Time step in the numerical integration.\n * \\`steps_per_frame\\`: How many time steps for each frame render. Useful to speed up the animation.\n * \\`delay\\`: A short pause between frames. Useful to slow down the animation.\n * \\`period\\`: Reset the initial conditions after a certain period. Useful to maintain stability of\n periodic orbits.\n * \\`cnt_max\\`: Stop the animation after some number of iterations. Useful for some configurations that explode. \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1265,"value":"$4d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1420,"value":"$4e","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1497,"value":"md`### Numerical integration\n\nThe system is solved using this [basic Runge-Kutta implementation](https://observablehq.com/@mcmcclur/runge-kutta-for-systems-of-odes). The following function accepts a list of nodes, sets up the equations of motion, and returns a function that integrates those equations using the parameters listed in the properties of nodes.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":245,"value":"$4f","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1462,"value":"md`### Generating random and circular configurations`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":612,"value":"function random_object(seed) {\n return {\n m: d3.randomUniform.source(d3.randomLcg(seed))(1, 10)(),\n x: d3.randomUniform.source(d3.randomLcg(seed / 2))(-4, 4)(),\n y: d3.randomUniform.source(d3.randomLcg(seed / 3))(-4, 4)(),\n vx: 0, //d3.randomUniform.source(d3.randomLcg(seed / 4))(-4, 4)(),\n vy: 0 //d3.randomUniform.source(d3.randomLcg(seed / 5))(-4, 4)()\n };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":614,"value":"function random_configuration(k, seed) {\n let nodes = d3.range(k).map((_, i) => random_object(seed / i));\n nodes.xmin = -8;\n nodes.xmax = 8;\n nodes.ymin = -8;\n nodes.ymax = 8;\n nodes.steps_per_frame = 1000;\n nodes.dt = 0.00005;\n nodes.delay = 5;\n nodes.cnt_max = 5000;\n return nodes;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":663,"value":"$50","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1459,"value":"md`## Imports`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1423,"value":"import { select } from \"@jashkenas/inputs\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1426,"value":"import { rk4_multi_step } from '@mcmcclur/runge-kutta-for-systems-of-odes'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1431,"value":"d3 = require('d3-selection@2', 'd3-array@2', 'd3-random@2', 'd3-shape@2', 'd3-scale@2', 'd3-scale-chromatic@2')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W51","stdlib":"1"}} 42:null