ring/dyson.html at main · bobjen/ring · GitHub

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<!DOCTYPE html>
<html>
<head>
  <title>Dyson 2-Ring Action Minimizer</title>
  <script src="ring.js"></script>
  <script src="action.js"></script>
</head>
<body bgcolor="#000000" text="#ffffff" style="font-family:monospace;">
<p><canvas id="cv" width="600" height="400" style="border:1px solid #ffffff;"></canvas></p>
<pre id="log" style="color:#00ff00;font-size:11px;max-height:300px;overflow:auto;"></pre>
<script>
(function() {
    var N   = 2;
    var rm  = 0.001;
    var r   = 0.4;
    var l   = 1 - r*r;
    var BASE = { increment: 0.1, work: 20, points: 8, trail: false };
    var inc  = BASE.increment / BASE.work;

    var colors = ['#ff0000', '#00ffff'];

    // ---- Orbit sampling (same approach as eight.html) ----

    function sampleOrbit(sunR) {
        var cosmos = new Cosmos2D(Object.assign({}, BASE, { moons: [
            { y: sunR, z: 0, vy: 0, vz: 0, mass: 1-N*rm, l: 0, fixed: true },
            { y: Math.sqrt(1 - r*r), z: r, l: l, vy: -r, vz: 0, mass: rm }
        ]}));
        var ring0 = cosmos.rings[1];
        var traj = [[ring0.p.y, ring0.p.z, ring0.v.y / inc, ring0.v.z / inc]];
        var prevVz = 0, wentNeg = false, wentPos = false, T = 0;
        for (var i = 0; i < 100000; i++) {
            cosmos.multistep();
            var vz = ring0.v.z;
            traj.push([ring0.p.y, ring0.p.z, ring0.v.y / inc, ring0.v.z / inc]);
            if (!wentNeg && vz < 0)              wentNeg = true;
            if ( wentNeg && !wentPos && vz >= 0) wentPos = true;
            if ( wentPos && vz < 0) {
                var alpha = prevVz / (prevVz - vz);
                T = i + alpha;
                break;
            }
            prevVz = vz;
        }
        var need = Math.ceil(2 * T) + 2;
        while (traj.length <= need) {
            cosmos.multistep();
            traj.push([ring0.p.y, ring0.p.z, ring0.v.y / inc, ring0.v.z / inc]);
        }
        var samples = [];
        for (var k = 0; k < N; k++) {
            var w  = k / N;
            var t1 = k * T / N,     i1 = Math.floor(t1), f1 = t1 - i1;
            var t2 = T + k * T / N, i2 = Math.floor(t2), f2 = t2 - i2;
            var p1 = traj[i1], q1 = traj[i1 + 1] || traj[i1];
            var p2 = traj[i2], q2 = traj[i2 + 1] || traj[i2];
            samples.push({
                y:  w*(p1[0]+f1*(q1[0]-p1[0])) + (1-w)*(p2[0]+f2*(q2[0]-p2[0])),
                z:  w*(p1[1]+f1*(q1[1]-p1[1])) + (1-w)*(p2[1]+f2*(q2[1]-p2[1])),
                vy: w*(p1[2]+f1*(q1[2]-p1[2])) + (1-w)*(p2[2]+f2*(q2[2]-p2[2])),
                vz: w*(p1[3]+f1*(q1[3]-p1[3])) + (1-w)*(p2[3]+f2*(q2[3]-p2[3])),
            });
        }
        return { samples: samples, T: T, traj: traj };
    }

    function makeMoons(samples, sunR) {
        var moons = [{ y: sunR, z: 0, vy: 0, vz: 0, mass: 1-N*rm, l: 0,
                       fixed: true, color: '#ffff00', radius: 10 }];
        for (var i = 0; i < N; i++)
            moons.push({ y: samples[i].y, z: samples[i].z,
                         vy: samples[i].vy, vz: samples[i].vz,
                         l: l, mass: rm, color: colors[i], radius: 3 });
        return moons;
    }

    // Run the full N-ring system for one period, re-sample.
    // Returns { samples, traj, T } so callers can also use the trajectory
    // (e.g. to initialise the action minimiser from the 2-ring orbit).
    function sampleOrbitFull(sunR, samples) {
        var cosmos = new Cosmos2D(Object.assign({}, BASE,
                                  { moons: makeMoons(samples, sunR) }));
        var ring0 = cosmos.rings[1];
        var traj = [[ring0.p.y, ring0.p.z, ring0.v.y / inc, ring0.v.z / inc]];
        var prevVz = 0, wentNeg = false, wentPos = false, T = 0;
        for (var i = 0; i < 100000; i++) {
            cosmos.multistep();
            var vz = ring0.v.z;
            traj.push([ring0.p.y, ring0.p.z, ring0.v.y / inc, ring0.v.z / inc]);
            if (!wentNeg && vz < 0)              wentNeg = true;
            if ( wentNeg && !wentPos && vz >= 0) wentPos = true;
            if ( wentPos && vz < 0) {
                var alpha = prevVz / (prevVz - vz);
                T = i + alpha;
                break;
            }
            prevVz = vz;
        }
        var need = Math.ceil(2 * T) + 2;
        while (traj.length <= need) {
            cosmos.multistep();
            traj.push([ring0.p.y, ring0.p.z, ring0.v.y / inc, ring0.v.z / inc]);
        }
        var out = [];
        for (var k = 0; k < N; k++) {
            var w  = k / N;
            var t1 = k * T / N,     i1 = Math.floor(t1), f1 = t1 - i1;
            var t2 = T + k * T / N, i2 = Math.floor(t2), f2 = t2 - i2;
            var p1 = traj[i1], q1 = traj[i1 + 1] || traj[i1];
            var p2 = traj[i2], q2 = traj[i2 + 1] || traj[i2];
            out.push({
                y:  w*(p1[0]+f1*(q1[0]-p1[0])) + (1-w)*(p2[0]+f2*(q2[0]-p2[0])),
                z:  w*(p1[1]+f1*(q1[1]-p1[1])) + (1-w)*(p2[1]+f2*(q2[1]-p2[1])),
                vy: w*(p1[2]+f1*(q1[2]-p1[2])) + (1-w)*(p2[2]+f2*(q2[2]-p2[2])),
                vz: w*(p1[3]+f1*(q1[3]-p1[3])) + (1-w)*(p2[3]+f2*(q2[3]-p2[3])),
            });
        }
        return { samples: out, traj: traj, T: T };
    }

    // Extract choreographic samples from the two z=0 crossings in a single-ring
    // trajectory.  For N=2, the time-reversal symmetry z(+d)=-z(-d) forces vy=0
    // at every z=0 crossing, so only (y, vz) need to be found.  The two crossings
    // (upward then downward) are separated by T/2 and serve as the starting states
    // for ring 0 and ring 1 respectively.  Returns null if crossings are not found.
    function z0Samples(traj, T) {
        var t0 = -1, y1 = 0, vz1 = 0;
        for (var i = 1; i < traj.length; i++) {
            if (traj[i-1][1] < 0 && traj[i][1] >= 0) {
                var alpha = -traj[i-1][1] / (traj[i][1] - traj[i-1][1]);
                y1  = traj[i-1][0] + alpha*(traj[i][0] - traj[i-1][0]);
                vz1 = traj[i-1][3] + alpha*(traj[i][3] - traj[i-1][3]);
                t0  = (i - 1) + alpha;
                break;
            }
        }
        if (t0 < 0) return null;
        // Find the downward crossing nearest to t0 + T/2.
        var tHalf = t0 + T / 2, best = -1, bestDist = T;
        for (var j = 1; j < traj.length - 1; j++) {
            if (traj[j][1] > 0 && traj[j+1][1] <= 0) {
                var dist = Math.abs(j - tHalf);
                if (dist < bestDist) { bestDist = dist; best = j; }
            }
        }
        if (best < 0) return null;
        var p = traj[best], q = traj[best + 1];
        var beta = p[1] / (p[1] - q[1]);
        return [
            { y: y1,                          z: 0, vy: 0, vz: vz1 },
            { y: p[0] + beta*(q[0] - p[0]),   z: 0, vy: 0,
              vz: p[3] + beta*(q[3] - p[3]) }
        ];
    }

    function halfOrbitDy(sunR) {
        var res = sampleOrbit(sunR);
        var samples = res.samples;
        var cosmos = new Cosmos2D(Object.assign({}, BASE,
                                  { moons: makeMoons(samples, sunR) }));
        var ring0 = cosmos.rings[1];
        var y0 = ring0.p.y, wentNeg = false, prevVz = 0;
        for (var i = 0; i < 50000; i++) {
            cosmos.multistep();
            var vz = ring0.v.z;
            if (!wentNeg && vz < -1e-12) wentNeg = true;
            if ( wentNeg && vz >= 0) {
                var alpha = -prevVz / (vz - prevVz);
                return ring0.p.y - (1 - alpha) * ring0.v.y - y0;
            }
            prevVz = vz;
        }
        return ring0.p.y - y0;
    }

    function findSunR(maxSun, nScan, eps) {
        var step = maxSun / nScan;
        var prev = { s: 0, dy: halfOrbitDy(0) };
        var sunR = 0;
        for (var i = 1; i <= nScan; i++) {
            var s  = i * step;
            var dy = halfOrbitDy(s);
            if (prev.dy * dy <= 0) {
                var lo = prev.s, dlo = prev.dy, hi = s;
                for (var j = 0; j < 80; j++) {
                    var mid = (lo + hi) * 0.5;
                    var dm  = halfOrbitDy(mid);
                    if (dlo * dm <= 0) { hi = mid; }
                    else               { lo = mid; dlo = dm; }
                    if (hi - lo < eps) break;
                }
                sunR = (lo + hi) * 0.5; break;
            }
            prev = { s: s, dy: dy };
        }
        return sunR;
    }

    // ---- Action minimizer ----
    // Refine the orbit to a true choreography using Fourier-space gradient descent.
    // traj/T come from sampleOrbitFull().  Returns AM samples, or null if diverged to NaN.
    //
    // Strategy: run the AM with all harmonics free (better convergence), then call
    // projectChoreography() to zero the n=k·N harmonics that create nonzero total
    // momentum.
    //
    // Current status: the AM does not yet converge to a good choreographic orbit for
    // N=2.  The physical 2-ring trajectory already has ~7% even harmonics, which
    // means the orbit lacks exact choreographic z-symmetry at these parameters.
    // The AM stalls at rms≈3e-2 rather than converging to near-zero.  The download
    // therefore uses whatever the AM produces (both rings on the same Fourier orbit)
    // but the dynamics will look wrong — this is a known open problem.

    function runActionMinimizer(traj, T) {
        var lg = document.getElementById('log');
        var am = new ActionMinimizer({
            N: N, rm: rm, l: l,
            fixed: [{ y: sunR, z: 0, mass: 1 - N*rm }],
        });
        am.trajToParams(traj, T, inc);
        lg.textContent += 'AM initial: T_phys=' + am.params.T_phys.toFixed(4) +
                          '  rms=' + am.step(0).toExponential(3) + '\n';
        am.run({
            nSteps:      2000,
            alpha:       1.0,
            eps:         1e-8,
            reportEvery: 500,
            onProgress:  function(iter, r, Tv) {
                lg.textContent += 'AM iter ' + iter + ': rms=' + r.toExponential(3) +
                                  '  T=' + Tv.toFixed(6) + '\n';
            },
        });
        // Zero harmonics divisible by N — enforces zero total momentum and COM
        // centred on the orbit.  See ActionMinimizer.projectChoreography().
        am.projectChoreography();
        var amSamples = am.getSamples();
        for (var k = 0; k < N; k++) {
            if (!isFinite(amSamples[k].y) || !isFinite(amSamples[k].z)) {
                lg.textContent += 'AM diverged to NaN — download uses physical samples.\n';
                return null;
            }
        }
        lg.textContent += 'AM Ring 0: y=' + amSamples[0].y.toFixed(6) + ' z=' + amSamples[0].z.toFixed(6) + '\n';
        lg.textContent += 'AM Ring 1: y=' + amSamples[1].y.toFixed(6) + ' z=' + amSamples[1].z.toFixed(6) + '\n';
        return amSamples;
    }

    // ---- Main ----
    var lg = document.getElementById('log');

    var sunR = findSunR(0.5, 20, 1e-6);
    lg.textContent += 'sunR=' + sunR.toFixed(8) + '\n';

    var res = sampleOrbit(sunR);
    // Seed from the z=0 crossings (upward then downward, separated by T/2).
    // By time-reversal symmetry vy=0 at both crossings, so the orbit starts in
    // the correct choreographic symmetry class and the iteration stays there.
    // Fall back to the t=0/t=T/2 samples if crossings are not found.
    var samples = z0Samples(res.traj, res.T) || res.samples;
    lg.textContent += 'z=0 seed: Ring 0: y=' + samples[0].y.toFixed(6) +
                      '  Ring 1: y=' + samples[1].y.toFixed(6) + '\n';

    // Refine to self-consistency under the full 2-ring system.
    // Keep the last full-system run so the AM can start from the 2-ring trajectory
    // instead of the single-ring one — the 2-ring orbit is already close to the
    // choreography, so the AM needs far smaller corrections.
    var fullRes = { traj: res.traj, T: res.T };
    for (var iter = 0; iter < 6; iter++) {
        var fr = sampleOrbitFull(sunR, samples);
        var maxD = 0;
        for (var k = 0; k < N; k++)
            maxD = Math.max(maxD, Math.abs(fr.samples[k].y - samples[k].y),
                                  Math.abs(fr.samples[k].z - samples[k].z));
        samples = fr.samples;
        fullRes = fr;
        if (maxD < 1e-9) break;
    }
    lg.textContent += 'Physical samples converged.\n';
    lg.textContent += 'Ring 0: y=' + samples[0].y.toFixed(6) + ' z=' + samples[0].z.toFixed(6) +
                      ' vz=' + samples[0].vz.toFixed(6) + '\n';
    lg.textContent += 'Ring 1: y=' + samples[1].y.toFixed(6) + ' z=' + samples[1].z.toFixed(6) +
                      ' vz=' + samples[1].vz.toFixed(6) + '\n';

    // Run action minimizer starting from the converged 2-ring trajectory.
    var amSamples = runActionMinimizer(fullRes.traj, fullRes.T);
    var downloadSamples = amSamples || samples;
    var downloadNote = amSamples ? '' : ' (action minimizer diverged — using physical samples)';

    var ringOpts = Object.assign({}, BASE, {
        background: '#000000',
        scale:    400,
        ymargin:  20,
        framerate: 20,
        trail:    true,
        traillen: 2000,
        fade:     0.02,
        stop:     false,
        moons:    makeMoons(samples, sunR),
    });

    ring('cv', ringOpts);

    // Download dyson_out.html using the action-minimized initial conditions.
    var a = document.createElement('a');
    a.textContent = 'Download dyson_out.html' + downloadNote;
    a.style.cssText = 'color:#00ffff;cursor:pointer;';
    a.onclick = function() {
        var downloadOpts = Object.assign({}, ringOpts, { moons: makeMoons(downloadSamples, sunR) });
        actionDownloadHTML('Dyson 2-Ring', downloadOpts);
    };
    document.body.appendChild(a);
})();
</script>
</body>
</html>