01Pendulum Stepped by Solving Both Arm Accelerations

<main class="stage">
<div class="bp is-demo" id="bpRoot">
  <canvas class="bp__canvas" id="bpCanvas" width="480" height="300" tabindex="0" aria-label="It solves both arm accelerations with the Lagrangian equations and integrates each frame in three Runge-Kutta steps, drawing the base pendulum whose tip follows the exact motion."></canvas>
  <p class="bp__status" id="bpStatus" aria-live="polite">Demo running</p>
  <button class="bp__btn" id="bpBtn" type="button">Pause</button>
  <p class="bp__caption">It solves both arm accelerations with the Lagrangian equations and integrates each frame in three Runge-Kutta steps, drawing the base pendulum whose tip follows the exact motion.</p>
</div>
</main>

@use "../../../../site/static/demo-tokens" as *;
@include stage($stage-ink);

.bp {
  position: relative;
  width: min(92vw, calc(160vh - 12.8vw));
  aspect-ratio: 480 / 300;
  margin: 0 auto;
  border-radius: $r-card;
  overflow: hidden;
  box-shadow: $shadow-raised;
  background: $stage-ink;
}

.bp__canvas {
  display: block;
  width: 100%;
  height: 100%;
  touch-action: none;
}

.bp__canvas:focus-visible {
  outline: 2px solid rgba(255, 255, 255, .85);
  outline-offset: -6px;
}

.bp__status {
  position: absolute;
  left: $sp-3;
  top: $sp-3;
  max-width: 70%;
  margin: 0;
  padding: $sp-1 $sp-3;
  border-radius: $r-pill;
  background: rgba(255, 255, 255, .85);
  color: $subject-ink;
  font-weight: 500;
  font-size: 11px;
  line-height: 1.4;
  pointer-events: none;
}

.bp__btn {
  position: absolute;
  right: $sp-3;
  bottom: $sp-3;
  margin: 0;
  padding: $sp-1 $sp-3;
  border: 0;
  border-radius: $r-pill;
  background: rgba(255, 255, 255, .85);
  color: $subject-ink;
  font: inherit;
  font-weight: 500;
  font-size: 12px;
  line-height: 1.4;
  cursor: pointer;
}

.bp__btn:focus-visible {
  outline: 2px solid #ffffff;
  outline-offset: 2px;
}

.bp__caption {
  position: absolute;
  left: $sp-3;
  bottom: $sp-3;
  max-width: 68%;
  margin: 0;
  padding: $sp-1 $sp-3;
  border-radius: $r-pill;
  background: rgba(255, 255, 255, .85);
  color: $subject-ink;
  font-weight: 500;
  font-size: 12px;
  line-height: 1.4;
  pointer-events: none;
}

@media (max-width: 340px) {
  .bp__caption, .bp__btn { font-size: 10px; padding: $sp-1 $sp-2; }
  .bp__status { font-size: 10px; }
}

@media (prefers-reduced-motion: reduce) {
  .bp { cursor: default; }
}

html, body {
  margin: 0;
  overflow: hidden;
}

.stage {
  box-sizing: border-box;
  min-height: 100vh;
  display: grid;
  place-items: center;
  margin: 0;
  background: #17141a;
  font-family: -apple-system, "Segoe UI", "Noto Sans KR", sans-serif;
  padding: clamp(8px, 4vw, 24px);
}

.bp {
  position: relative;
  width: min(92vw, 160vh - 12.8vw);
  aspect-ratio: 480/300;
  margin: 0 auto;
  border-radius: 16px;
  overflow: hidden;
  box-shadow: 0 14px 24px rgba(23, 20, 26, 0.32);
  background: #17141a;
}

.bp__canvas {
  display: block;
  width: 100%;
  height: 100%;
  touch-action: none;
}

.bp__canvas:focus-visible {
  outline: 2px solid rgba(255, 255, 255, 0.85);
  outline-offset: -6px;
}

.bp__status {
  position: absolute;
  left: 12px;
  top: 12px;
  max-width: 70%;
  margin: 0;
  padding: 4px 12px;
  border-radius: 999px;
  background: rgba(255, 255, 255, 0.85);
  color: #17141a;
  font-weight: 500;
  font-size: 11px;
  line-height: 1.4;
  pointer-events: none;
}

.bp__btn {
  position: absolute;
  right: 12px;
  bottom: 12px;
  margin: 0;
  padding: 4px 12px;
  border: 0;
  border-radius: 999px;
  background: rgba(255, 255, 255, 0.85);
  color: #17141a;
  font: inherit;
  font-weight: 500;
  font-size: 12px;
  line-height: 1.4;
  cursor: pointer;
}

.bp__btn:focus-visible {
  outline: 2px solid #ffffff;
  outline-offset: 2px;
}

.bp__caption {
  position: absolute;
  left: 12px;
  bottom: 12px;
  max-width: 68%;
  margin: 0;
  padding: 4px 12px;
  border-radius: 999px;
  background: rgba(255, 255, 255, 0.85);
  color: #17141a;
  font-weight: 500;
  font-size: 12px;
  line-height: 1.4;
  pointer-events: none;
}

@media (max-width: 340px) {
  .bp__caption, .bp__btn {
    font-size: 10px;
    padding: 4px 8px;
  }
  .bp__status {
    font-size: 10px;
  }
}
@media (prefers-reduced-motion: reduce) {
  .bp {
    cursor: default;
  }
}

(function () {
  var root = document.getElementById('bpRoot');
  var canvas = document.getElementById('bpCanvas');
  var statusEl = document.getElementById('bpStatus');
  var btnEl = document.getElementById('bpBtn');
  "use strict";
  function main(root, canvas, statusEl, btnEl) {
      var ctx0 = canvas.getContext('2d');
      if (!ctx0) {
          return function () { return; };
      }
      var ctx = ctx0;
      var wrap = canvas.parentElement ? canvas.parentElement : root;
      var CALC_W = 320, CALC_H = 200;
      var FIXED_DT = 16;
      var REDUCE_STEP_FRAMES = 40;
      var reduce = window.matchMedia('(prefers-reduced-motion: reduce)').matches;
      var demo = !reduce;
      if (!demo)
          root.classList.remove('is-demo');
      function stopDemo() { demo = false; root.classList.remove('is-demo'); }
      var paused = false;
      var ownInput = false;
      var cw = 480, ch = 300, sc = 1.5, elapsed = 0, rafId = 0;
      var cleaners = [];
      function listen(target, type, fn) {
          target.addEventListener(type, fn);
          cleaners.push(function () { target.removeEventListener(type, fn); });
      }
      function clamp(v, lo, hi) { return v < lo ? lo : (v > hi ? hi : v); }
      function setStatus(msg) { statusEl.textContent = msg; }
      function hsl(h, s, l, a) {
          return 'hsla(' + (h % 360) + ',' + s + '%,' + l + '%,' + a + ')';
      }
      function canvasPoint(e) {
          var r = canvas.getBoundingClientRect();
          return { x: (e.clientX - r.left) / r.width * CALC_W, y: (e.clientY - r.top) / r.height * CALC_H };
      }
      // 배경 — 어두운 잉크색 위에 방사형 빛무리를 (gx, gy) 에 깐다
      function beginDraw(inner, outer, gx, gy) {
          ctx.setTransform(1, 0, 0, 1, 0, 0);
          ctx.fillStyle = '#1d1a2b';
          ctx.fillRect(0, 0, cw, ch);
          ctx.setTransform(sc, 0, 0, sc, 0, 0);
          var g = ctx.createRadialGradient(gx, gy, 8, gx, gy, 230);
          g.addColorStop(0, inner);
          g.addColorStop(1, outer);
          ctx.fillStyle = g;
          ctx.fillRect(0, 0, CALC_W, CALC_H);
      }
      function seg(x1, y1, x2, y2, w, col) {
          ctx.beginPath();
          ctx.moveTo(x1, y1);
          ctx.lineTo(x2, y2);
          ctx.lineCap = 'round';
          ctx.lineWidth = w;
          ctx.strokeStyle = col;
          ctx.stroke();
      }
      function disc(x, y, r, col) {
          ctx.beginPath();
          ctx.arc(x, y, r, 0, 6.2832);
          ctx.fillStyle = col;
          ctx.fill();
      }
      function glow(x, y, r, rgb, a) {
          var g = ctx.createRadialGradient(x, y, 0, x, y, r);
          g.addColorStop(0, 'rgba(' + rgb + ',' + a + ')');
          g.addColorStop(1, 'rgba(' + rgb + ',0)');
          ctx.fillStyle = g;
          ctx.fillRect(x - r, y - r, r * 2, r * 2);
      }
      function drawTrail(arr, rgb, w, alpha) {
          ctx.lineCap = 'round';
          for (var i = 1; i < arr.length; i += 1) {
              var f = i / arr.length;
              ctx.beginPath();
              ctx.moveTo(arr[i - 1].x, arr[i - 1].y);
              ctx.lineTo(arr[i].x, arr[i].y);
              ctx.strokeStyle = 'rgba(' + rgb + ',' + (alpha * f * f) + ')';
              ctx.lineWidth = w * (0.3 + 0.7 * f);
              ctx.stroke();
          }
      }
      function pushTrail(arr, x, y, max) {
          arr.push({ x: x, y: y });
          if (arr.length > max) {
              arr.shift();
          }
      }
      function defaultToggle() {
          if (reduce) {
              for (var i = 0; i < REDUCE_STEP_FRAMES; i += 1) {
                  elapsed += FIXED_DT;
                  stepFrame();
              }
              draw();
              setStatus('Stepping one scene at a time');
              return;
          }
          paused = !paused;
          btnEl.textContent = paused ? 'Play' : 'Pause';
          setStatus(paused ? 'Paused' : 'Demo running');
      }
      var onButton = defaultToggle;
      // 기법: 두 각도의 가속도를 라그랑주 식으로 풀고, 한 프레임을 SUBSTEPS 걸음으로 쪼개 룽게-쿠타로 적분한다
      var DT = 0.01;
      var SUBSTEPS = 3;
      var START_T1 = 2.2;
      var START_T2 = 2.6;
      var TRAIL_LEN = 72;
      var L1 = 1.0;
      var L2 = 1.0;
      var M1 = 1.0;
      var M2 = 1.0;
      var G = 9.81;
      var UNIT = 38, PIVOT_X = 160, PIVOT_Y = 74;
      var PREROLL_FRAMES = 30;
      // 라그랑주 운동방정식 — 각도는 아래 방향이 0, 시계 방향이 양수
      function accTheta1(t1, t2, w1, w2) {
          var d = t1 - t2;
          var den = L1 * (2 * M1 + M2 - M2 * Math.cos(2 * d));
          return (-G * (2 * M1 + M2) * Math.sin(t1) - M2 * G * Math.sin(t1 - 2 * t2)
              - 2 * Math.sin(d) * M2 * (w2 * w2 * L2 + w1 * w1 * L1 * Math.cos(d))) / den;
      }
      function accTheta2(t1, t2, w1, w2) {
          var d = t1 - t2;
          var den = L2 * (2 * M1 + M2 - M2 * Math.cos(2 * d));
          return (2 * Math.sin(d) * (w1 * w1 * L1 * (M1 + M2) + G * (M1 + M2) * Math.cos(t1)
              + w2 * w2 * L2 * M2 * Math.cos(d))) / den;
      }
      // 룽게-쿠타 4단계 — 한 걸음을 네 번 기울기로 평균 낸다 (s = [t1, t2, w1, w2])
      function rk4Step(s, h) {
          var t1 = s[0], t2 = s[1], w1 = s[2], w2 = s[3];
          var h2 = h * 0.5;
          var k1a = w1, k1b = w2, k1c = accTheta1(t1, t2, w1, w2), k1d = accTheta2(t1, t2, w1, w2);
          var u1 = t1 + h2 * k1a, u2 = t2 + h2 * k1b, u3 = w1 + h2 * k1c, u4 = w2 + h2 * k1d;
          var k2a = u3, k2b = u4, k2c = accTheta1(u1, u2, u3, u4), k2d = accTheta2(u1, u2, u3, u4);
          u1 = t1 + h2 * k2a;
          u2 = t2 + h2 * k2b;
          u3 = w1 + h2 * k2c;
          u4 = w2 + h2 * k2d;
          var k3a = u3, k3b = u4, k3c = accTheta1(u1, u2, u3, u4), k3d = accTheta2(u1, u2, u3, u4);
          u1 = t1 + h * k3a;
          u2 = t2 + h * k3b;
          u3 = w1 + h * k3c;
          u4 = w2 + h * k3d;
          var k4a = u3, k4b = u4, k4c = accTheta1(u1, u2, u3, u4), k4d = accTheta2(u1, u2, u3, u4);
          s[0] = t1 + h / 6 * (k1a + 2 * k2a + 2 * k3a + k4a);
          s[1] = t2 + h / 6 * (k1b + 2 * k2b + 2 * k3b + k4b);
          s[2] = w1 + h / 6 * (k1c + 2 * k2c + 2 * k3c + k4c);
          s[3] = w2 + h / 6 * (k1d + 2 * k2d + 2 * k3d + k4d);
      }
      function bobOne(s, ox, oy) {
          return { x: ox + UNIT * L1 * Math.sin(s[0]), y: oy + UNIT * L1 * Math.cos(s[0]) };
      }
      function bobTwo(s, ox, oy) {
          var a = bobOne(s, ox, oy);
          return { x: a.x + UNIT * L2 * Math.sin(s[1]), y: a.y + UNIT * L2 * Math.cos(s[1]) };
      }
      function drawArms(s, ox, oy, c1, c2, w) {
          var a = bobOne(s, ox, oy), b = bobTwo(s, ox, oy);
          seg(ox, oy, a.x, a.y, w, c1);
          seg(a.x, a.y, b.x, b.y, w, c2);
      }
      var sim = [START_T1, START_T2, 0, 0];
      var accA = 0, accB = 0;
      var trail = [];
      function stepFrame() {
          for (var i = 0; i < SUBSTEPS; i += 1) {
              rk4Step(sim, DT);
          }
          accA = accTheta1(sim[0], sim[1], sim[2], sim[3]);
          accB = accTheta2(sim[0], sim[1], sim[2], sim[3]);
          var p = bobTwo(sim, PIVOT_X, PIVOT_Y);
          pushTrail(trail, p.x, p.y, TRAIL_LEN);
      }
      function meter(y, v, rgb) {
          var half = 52;
          ctx.fillStyle = 'rgba(255,255,255,0.14)';
          ctx.fillRect(14, y, half * 2, 6);
          ctx.fillStyle = 'rgba(' + rgb + ',0.95)';
          var f = clamp(v / 90, -1, 1) * half;
          ctx.fillRect(14 + half + (f < 0 ? f : 0), y, Math.abs(f), 6);
          ctx.fillStyle = 'rgba(255,255,255,0.7)';
          ctx.fillRect(14 + half - 0.5, y - 2, 1, 10);
      }
      function draw() {
          var a = bobOne(sim, PIVOT_X, PIVOT_Y), b = bobTwo(sim, PIVOT_X, PIVOT_Y);
          beginDraw('#4a416e', '#28223f', b.x, b.y);
          glow(b.x, b.y, 96, '255,238,252', 0.3);
          ctx.strokeStyle = 'rgba(255,255,255,0.12)';
          ctx.lineWidth = 1;
          ctx.setLineDash([3, 5]);
          ctx.beginPath();
          ctx.arc(PIVOT_X, PIVOT_Y, UNIT * (L1 + L2), 0, 6.2832);
          ctx.stroke();
          ctx.beginPath();
          ctx.arc(PIVOT_X, PIVOT_Y, UNIT * L1, 0, 6.2832);
          ctx.stroke();
          ctx.setLineDash([]);
          drawTrail(trail, '255,179,199', 4, 0.8);
          drawArms(sim, PIVOT_X, PIVOT_Y, '#c9b8ff', '#9fd8ff', 5);
          seg(PIVOT_X, PIVOT_Y, a.x, a.y, 1.5, 'rgba(255,255,255,0.55)');
          seg(a.x, a.y, b.x, b.y, 1.5, 'rgba(255,255,255,0.55)');
          disc(PIVOT_X, PIVOT_Y, 5, '#fff1c9');
          glow(a.x, a.y, 20, '201,184,255', 0.55);
          disc(a.x, a.y, 8, '#c9b8ff');
          glow(b.x, b.y, 26, '255,179,199', 0.65);
          disc(b.x, b.y, 10, '#ffb3c7');
          disc(b.x - 3, b.y - 3, 3, 'rgba(255,255,255,0.8)');
          meter(30, accA, '201,184,255');
          meter(42, accB, '159,216,255');
      }
      function fit() {
          var r = wrap.getBoundingClientRect();
          cw = Math.max(1, Math.round(r.width));
          ch = Math.max(1, Math.round(r.height));
          canvas.width = cw;
          canvas.height = ch;
          sc = cw / CALC_W;
          draw();
      }
      var lastTs = null;
      var accMs = 0;
      function tick(ts) {
          if (lastTs === null) {
              lastTs = ts;
          }
          var frameDt = Math.min(ts - lastTs, 48);
          lastTs = ts;
          if (!paused) {
              accMs += frameDt;
              while (accMs >= FIXED_DT) {
                  elapsed += FIXED_DT;
                  stepFrame();
                  accMs -= FIXED_DT;
              }
          }
          draw();
          rafId = requestAnimationFrame(tick);
      }
      for (var w0 = 0; w0 < PREROLL_FRAMES; w0 += 1) {
          elapsed += FIXED_DT;
          stepFrame();
      }
      fit();
      listen(window, 'resize', fit);
      listen(btnEl, 'click', function () { onButton(); });
      if (!ownInput) {
          listen(canvas, 'pointerdown', function () { onButton(); });
          listen(canvas, 'keydown', function (e) {
              var k = e.key;
              if (k === 'Enter' || k === ' ') {
                  e.preventDefault();
                  onButton();
              }
          });
      }
      if (reduce) {
          if (!ownInput) {
              btnEl.textContent = 'Next scene';
              setStatus('Stepping one scene at a time');
          }
      }
      else {
          rafId = requestAnimationFrame(tick);
      }
      function destroy() {
          cancelAnimationFrame(rafId);
          for (var i = 0; i < cleaners.length; i += 1) {
              cleaners[i]();
          }
      }
      return destroy;
  }
  main(root, canvas, statusEl, btnEl);
})();

import { useEffect, useRef } from "react";
import styles from "./DpRk4TwoArmStepLagrangian.module.scss";

function main(root: HTMLDivElement, canvas: HTMLCanvasElement, statusEl: HTMLElement, btnEl: HTMLButtonElement): () => void {
  type Pt = { x: number; y: number };
  var ctx0 = canvas.getContext('2d');
  if (!ctx0) { return function (): void { return; }; }
  var ctx: CanvasRenderingContext2D = ctx0;
  var wrap: HTMLElement = canvas.parentElement ? canvas.parentElement : root;
  var CALC_W = 320, CALC_H = 200;
  var FIXED_DT = 16;
  var REDUCE_STEP_FRAMES = 40;
  var reduce = window.matchMedia('(prefers-reduced-motion: reduce)').matches;
  var demo = !reduce;
  if (!demo) root.classList.remove('is-demo');
  function stopDemo(): void { demo = false; root.classList.remove('is-demo'); }
  var paused = false;
  var ownInput = false;
  var cw = 480, ch = 300, sc = 1.5, elapsed = 0, rafId = 0;
  var cleaners: Array<() => void> = [];

  function listen(target: EventTarget, type: string, fn: (e: Event) => void): void {
    target.addEventListener(type, fn);
    cleaners.push(function (): void { target.removeEventListener(type, fn); });
  }
  function clamp(v: number, lo: number, hi: number): number { return v < lo ? lo : (v > hi ? hi : v); }
  function setStatus(msg: string): void { statusEl.textContent = msg; }
  function hsl(h: number, s: number, l: number, a: number): string {
    return 'hsla(' + (h % 360) + ',' + s + '%,' + l + '%,' + a + ')';
  }
  function canvasPoint(e: PointerEvent): Pt {
    var r = canvas.getBoundingClientRect();
    return { x: (e.clientX - r.left) / r.width * CALC_W, y: (e.clientY - r.top) / r.height * CALC_H };
  }

  // 배경 — 어두운 잉크색 위에 방사형 빛무리를 (gx, gy) 에 깐다
  function beginDraw(inner: string, outer: string, gx: number, gy: number): void {
    ctx.setTransform(1, 0, 0, 1, 0, 0);
    ctx.fillStyle = '#1d1a2b';
    ctx.fillRect(0, 0, cw, ch);
    ctx.setTransform(sc, 0, 0, sc, 0, 0);
    var g = ctx.createRadialGradient(gx, gy, 8, gx, gy, 230);
    g.addColorStop(0, inner);
    g.addColorStop(1, outer);
    ctx.fillStyle = g;
    ctx.fillRect(0, 0, CALC_W, CALC_H);
  }
  function seg(x1: number, y1: number, x2: number, y2: number, w: number, col: string): void {
    ctx.beginPath();
    ctx.moveTo(x1, y1);
    ctx.lineTo(x2, y2);
    ctx.lineCap = 'round';
    ctx.lineWidth = w;
    ctx.strokeStyle = col;
    ctx.stroke();
  }
  function disc(x: number, y: number, r: number, col: string): void {
    ctx.beginPath();
    ctx.arc(x, y, r, 0, 6.2832);
    ctx.fillStyle = col;
    ctx.fill();
  }
  function glow(x: number, y: number, r: number, rgb: string, a: number): void {
    var g = ctx.createRadialGradient(x, y, 0, x, y, r);
    g.addColorStop(0, 'rgba(' + rgb + ',' + a + ')');
    g.addColorStop(1, 'rgba(' + rgb + ',0)');
    ctx.fillStyle = g;
    ctx.fillRect(x - r, y - r, r * 2, r * 2);
  }
  function drawTrail(arr: Pt[], rgb: string, w: number, alpha: number): void {
    ctx.lineCap = 'round';
    for (var i = 1; i < arr.length; i += 1) {
      var f = i / arr.length;
      ctx.beginPath();
      ctx.moveTo(arr[i - 1].x, arr[i - 1].y);
      ctx.lineTo(arr[i].x, arr[i].y);
      ctx.strokeStyle = 'rgba(' + rgb + ',' + (alpha * f * f) + ')';
      ctx.lineWidth = w * (0.3 + 0.7 * f);
      ctx.stroke();
    }
  }
  function pushTrail(arr: Pt[], x: number, y: number, max: number): void {
    arr.push({ x: x, y: y });
    if (arr.length > max) { arr.shift(); }
  }

  function defaultToggle(): void {
    if (reduce) {
      for (var i = 0; i < REDUCE_STEP_FRAMES; i += 1) { elapsed += FIXED_DT; stepFrame(); }
      draw();
      setStatus('한 장면씩 넘기는 중');
      return;
    }
    paused = !paused;
    btnEl.textContent = paused ? '재생' : '멈춤';
    setStatus(paused ? '일시정지됨' : '시연 중');
  }
  var onButton: () => void = defaultToggle;
  // 기법: 두 각도의 가속도를 라그랑주 식으로 풀고, 한 프레임을 SUBSTEPS 걸음으로 쪼개 룽게-쿠타로 적분한다
  var DT = 0.01;
  var SUBSTEPS = 3;
  var START_T1 = 2.2;
  var START_T2 = 2.6;
  var TRAIL_LEN = 72;
  var L1 = 1.0;
  var L2 = 1.0;
  var M1 = 1.0;
  var M2 = 1.0;
  var G = 9.81;
  var UNIT = 38, PIVOT_X = 160, PIVOT_Y = 74;
  var PREROLL_FRAMES = 30;
  // 라그랑주 운동방정식 — 각도는 아래 방향이 0, 시계 방향이 양수
  function accTheta1(t1: number, t2: number, w1: number, w2: number): number {
    var d = t1 - t2;
    var den = L1 * (2 * M1 + M2 - M2 * Math.cos(2 * d));
    return (-G * (2 * M1 + M2) * Math.sin(t1) - M2 * G * Math.sin(t1 - 2 * t2)
      - 2 * Math.sin(d) * M2 * (w2 * w2 * L2 + w1 * w1 * L1 * Math.cos(d))) / den;
  }
  function accTheta2(t1: number, t2: number, w1: number, w2: number): number {
    var d = t1 - t2;
    var den = L2 * (2 * M1 + M2 - M2 * Math.cos(2 * d));
    return (2 * Math.sin(d) * (w1 * w1 * L1 * (M1 + M2) + G * (M1 + M2) * Math.cos(t1)
      + w2 * w2 * L2 * M2 * Math.cos(d))) / den;
  }
  // 룽게-쿠타 4단계 — 한 걸음을 네 번 기울기로 평균 낸다 (s = [t1, t2, w1, w2])
  function rk4Step(s: number[], h: number): void {
    var t1 = s[0], t2 = s[1], w1 = s[2], w2 = s[3];
    var h2 = h * 0.5;
    var k1a = w1, k1b = w2, k1c = accTheta1(t1, t2, w1, w2), k1d = accTheta2(t1, t2, w1, w2);
    var u1 = t1 + h2 * k1a, u2 = t2 + h2 * k1b, u3 = w1 + h2 * k1c, u4 = w2 + h2 * k1d;
    var k2a = u3, k2b = u4, k2c = accTheta1(u1, u2, u3, u4), k2d = accTheta2(u1, u2, u3, u4);
    u1 = t1 + h2 * k2a; u2 = t2 + h2 * k2b; u3 = w1 + h2 * k2c; u4 = w2 + h2 * k2d;
    var k3a = u3, k3b = u4, k3c = accTheta1(u1, u2, u3, u4), k3d = accTheta2(u1, u2, u3, u4);
    u1 = t1 + h * k3a; u2 = t2 + h * k3b; u3 = w1 + h * k3c; u4 = w2 + h * k3d;
    var k4a = u3, k4b = u4, k4c = accTheta1(u1, u2, u3, u4), k4d = accTheta2(u1, u2, u3, u4);
    s[0] = t1 + h / 6 * (k1a + 2 * k2a + 2 * k3a + k4a);
    s[1] = t2 + h / 6 * (k1b + 2 * k2b + 2 * k3b + k4b);
    s[2] = w1 + h / 6 * (k1c + 2 * k2c + 2 * k3c + k4c);
    s[3] = w2 + h / 6 * (k1d + 2 * k2d + 2 * k3d + k4d);
  }
  function bobOne(s: number[], ox: number, oy: number): Pt {
    return { x: ox + UNIT * L1 * Math.sin(s[0]), y: oy + UNIT * L1 * Math.cos(s[0]) };
  }
  function bobTwo(s: number[], ox: number, oy: number): Pt {
    var a = bobOne(s, ox, oy);
    return { x: a.x + UNIT * L2 * Math.sin(s[1]), y: a.y + UNIT * L2 * Math.cos(s[1]) };
  }
  function drawArms(s: number[], ox: number, oy: number, c1: string, c2: string, w: number): void {
    var a = bobOne(s, ox, oy), b = bobTwo(s, ox, oy);
    seg(ox, oy, a.x, a.y, w, c1);
    seg(a.x, a.y, b.x, b.y, w, c2);
  }

  var sim: number[] = [START_T1, START_T2, 0, 0];
  var accA = 0, accB = 0;
  var trail: Pt[] = [];

  function stepFrame(): void {
    for (var i = 0; i < SUBSTEPS; i += 1) { rk4Step(sim, DT); }
    accA = accTheta1(sim[0], sim[1], sim[2], sim[3]);
    accB = accTheta2(sim[0], sim[1], sim[2], sim[3]);
    var p = bobTwo(sim, PIVOT_X, PIVOT_Y);
    pushTrail(trail, p.x, p.y, TRAIL_LEN);
  }

  function meter(y: number, v: number, rgb: string): void {
    var half = 52;
    ctx.fillStyle = 'rgba(255,255,255,0.14)';
    ctx.fillRect(14, y, half * 2, 6);
    ctx.fillStyle = 'rgba(' + rgb + ',0.95)';
    var f = clamp(v / 90, -1, 1) * half;
    ctx.fillRect(14 + half + (f < 0 ? f : 0), y, Math.abs(f), 6);
    ctx.fillStyle = 'rgba(255,255,255,0.7)';
    ctx.fillRect(14 + half - 0.5, y - 2, 1, 10);
  }

  function draw(): void {
    var a = bobOne(sim, PIVOT_X, PIVOT_Y), b = bobTwo(sim, PIVOT_X, PIVOT_Y);
    beginDraw('#4a416e', '#28223f', b.x, b.y);
    glow(b.x, b.y, 96, '255,238,252', 0.3);
    ctx.strokeStyle = 'rgba(255,255,255,0.12)';
    ctx.lineWidth = 1;
    ctx.setLineDash([3, 5]);
    ctx.beginPath();
    ctx.arc(PIVOT_X, PIVOT_Y, UNIT * (L1 + L2), 0, 6.2832);
    ctx.stroke();
    ctx.beginPath();
    ctx.arc(PIVOT_X, PIVOT_Y, UNIT * L1, 0, 6.2832);
    ctx.stroke();
    ctx.setLineDash([]);
    drawTrail(trail, '255,179,199', 4, 0.8);
    drawArms(sim, PIVOT_X, PIVOT_Y, '#c9b8ff', '#9fd8ff', 5);
    seg(PIVOT_X, PIVOT_Y, a.x, a.y, 1.5, 'rgba(255,255,255,0.55)');
    seg(a.x, a.y, b.x, b.y, 1.5, 'rgba(255,255,255,0.55)');
    disc(PIVOT_X, PIVOT_Y, 5, '#fff1c9');
    glow(a.x, a.y, 20, '201,184,255', 0.55);
    disc(a.x, a.y, 8, '#c9b8ff');
    glow(b.x, b.y, 26, '255,179,199', 0.65);
    disc(b.x, b.y, 10, '#ffb3c7');
    disc(b.x - 3, b.y - 3, 3, 'rgba(255,255,255,0.8)');
    meter(30, accA, '201,184,255');
    meter(42, accB, '159,216,255');
  }

  function fit(): void {
    var r = wrap.getBoundingClientRect();
    cw = Math.max(1, Math.round(r.width));
    ch = Math.max(1, Math.round(r.height));
    canvas.width = cw;
    canvas.height = ch;
    sc = cw / CALC_W;
    draw();
  }
  var lastTs: number | null = null;
  var accMs = 0;
  function tick(ts: number): void {
    if (lastTs === null) { lastTs = ts; }
    var frameDt = Math.min(ts - lastTs, 48);
    lastTs = ts;
    if (!paused) {
      accMs += frameDt;
      while (accMs >= FIXED_DT) { elapsed += FIXED_DT; stepFrame(); accMs -= FIXED_DT; }
    }
    draw();
    rafId = requestAnimationFrame(tick);
  }
  for (var w0 = 0; w0 < PREROLL_FRAMES; w0 += 1) { elapsed += FIXED_DT; stepFrame(); }
  fit();
  listen(window, 'resize', fit);
  listen(btnEl, 'click', function (): void { onButton(); });
  if (!ownInput) {
    listen(canvas, 'pointerdown', function (): void { onButton(); });
    listen(canvas, 'keydown', function (e: Event): void {
      var k = (e as KeyboardEvent).key;
      if (k === 'Enter' || k === ' ') { e.preventDefault(); onButton(); }
    });
  }
  if (reduce) {
    if (!ownInput) { btnEl.textContent = '다음 장면'; setStatus('한 장면씩 넘기는 중'); }
  } else {
    rafId = requestAnimationFrame(tick);
  }

  function destroy(): void {
    cancelAnimationFrame(rafId);
    for (var i = 0; i < cleaners.length; i += 1) { cleaners[i](); }
  }
  return destroy;
}


export default function DpRk4TwoArmStepLagrangian() {
  const rootRef = useRef<HTMLDivElement | null>(null);
  const canvasRef = useRef<HTMLCanvasElement | null>(null);
  const statusRef = useRef<HTMLParagraphElement | null>(null);
  const btnRef = useRef<HTMLButtonElement | null>(null);

  useEffect(() => {
    const root = rootRef.current;
    const canvas = canvasRef.current;
    const statusEl = statusRef.current;
    const btnEl = btnRef.current;
    if (!root || !canvas || !statusEl || !btnEl) return;
    return main(root, canvas, statusEl, btnEl);
  }, []);

  return (
    <main className={styles.stage}>
      <div className={`${styles.bp} is-demo`} ref={rootRef}>
        <canvas className={styles.bp__canvas} ref={canvasRef} width={480} height={300} tabIndex={0} aria-label="두 팔의 각가속도를 라그랑주 식으로 풀고 룽게-쿠타 네 단계로 한 프레임을 세 걸음에 나눠 적분해, 끝 추가가 정확히 따라오는 기본 진자를 그린다." />
        <p className={styles.bp__status} ref={statusRef} aria-live="polite">시연 중</p>
        <button className={styles.bp__btn} ref={btnRef} type="button">멈춤</button>
        <p className={styles.bp__caption}>두 팔의 각가속도를 라그랑주 식으로 풀고 룽게-쿠타 네 단계로 한 프레임을 세 걸음에 나눠 적분해, 끝 추가가 정확히 따라오는 기본 진자를 그린다.</p>
      </div>
    </main>
  );
}