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Hinweis: Mit dem Regler "Dämpfung" kann eingestellt werden, ob sich die Bewegung aufschaukelt (niedrige Dämpfung) oder immer weiter abschwächt (große Dämpfung)
[[Doppelpendel]]
<jsxgraph box="box" width="500" height="500">

  const board = JXG.JSXGraph.initBoard('box', {
    boundingbox: [-4, 4, 4, -4],
    axis: false,
    showCopyright: false,
    showNavigation: false
  });

  //Zufallszah
  var r = Math.random();                    // Zahl in [0,1)
  var formatted = r.toFixed(3);    

  // Parameter des Doppelpendels
  const g = 9.81;   // Erdbeschleunigung
  const l1 = 2;   // Länge erstes Pendel
  const l2 = 2;   // Länge zweites Pendel
  const m1 = 2.0;   // Masse 1
  const m2 = 1.0;   // Masse 2

  // Anfangsbedingungen
  let theta1 = r*Math.PI/2;
  let theta2 = r*Math.PI/2;
  let omega1 = r;
  let omega2 = r;

  
  let daempf = board.create('slider', [[-2,2.5],[2,2.5],[4, 4.4, 7]], {name:'Dämpfung',snapWidth:0.1});

  let running = true;   // Flag für Animation

  // Hilfsfunktionen für Differentialgleichungen
  function dtheta1() { return omega1; }
  function dtheta2() { return omega2; }

  function domega1() {
    const num1 = -g*(2*m1+m2)*Math.sin(theta1);
    const num2 = -m2*g*Math.sin(theta1-2*theta2);
    const num3 = -2*Math.sin(theta1-theta2)*m2*(omega2*omega2*l2 + omega1*omega1*l1*Math.cos(theta1-theta2));
    const den = l1*(2*m1+m2 - m2*Math.cos(2*theta1-2*theta2));
    return (num1 + num2 + num3) / den;
  }

  function domega2() {
    const num1 = 2*Math.sin(theta1-theta2);
    const num2 = omega1*omega1*l1*(m1+m2);
    const num3 = g*(m1+m2)*Math.cos(theta1);
    const num4 = omega2*omega2*l2*m2*Math.cos(theta1-theta2);
    const den = l2*(2*m1+m2 - m2*Math.cos(2*theta1-2*theta2));
    return (num1*(num2+num3+num4)) / den;
  }

  // Punkte im Board
  const origin = board.create('point', [0,0], {fixed:true, visible:false});
  const p1 = board.create('point', [
    () => l1*Math.sin(theta1),
    () => -l1*Math.cos(theta1)
  ], {name:'', size:2, face:'o', fixed:true});
     let p2 = board.create('point', [
      () => l1*Math.sin(theta1) + l2*Math.sin(theta2),
      () => -l1*Math.cos(theta1) - l2*Math.cos(theta2)
    ], {size:2, name:'', color: 'blue'});
    let p3 = board.create('point', [
      () => l1*Math.sin(theta1) + l2*Math.sin(theta2),
      () => -l1*Math.cos(theta1) - l2*Math.cos(theta2)
    ], {size:1, name:'', fillColor: 'rgba(0, 0, 255, 0.1)',strokeColor: 'rgba(255, 0, 0, 0)', trace:true});

  // Stäbe
  board.create('line', [origin, p1], {straightFirst:false, straightLast:false, strokeWidth:2});
  board.create('line', [p1, p2], {straightFirst:false, straightLast:false, strokeWidth:2});

  // Update-Schleife
  function step() {
    if (running) {
      const dt = 0.03; // Zeitschritt

      // Runge-Kutta 4
      let k1_theta1 = dtheta1();
      let k1_theta2 = dtheta2();
      let k1_omega1 = domega1();
      let k1_omega2 = domega2();

      let _theta1 = theta1 + dt*k1_theta1;
      let _theta2 = theta2 + dt*k1_theta2;
      let _omega1 = omega1 + dt*k1_omega1;
      let _omega2 = omega2 + dt*k1_omega2;

      let k2_theta1 = _omega1;
      let k2_theta2 = _omega2;
      let k2_omega1, k2_omega2;

      {
        let save_t1 = theta1, save_t2 = theta2, save_o1 = omega1, save_o2 = omega2;
        theta1 = _theta1; theta2 = _theta2; omega1 = _omega1; omega2 = _omega2;
        k2_omega1 = domega1();
        k2_omega2 = domega2();
        theta1 = save_t1; theta2 = save_t2; omega1 = save_o1; omega2 = save_o2;
      }

      _theta1 = theta1 + 0.5*dt*k2_theta1;
      _theta2 = theta2 + 0.5*dt*k2_theta2;
      _omega1 = omega1 + 0.5*dt*k2_omega1;
      _omega2 = omega2 + 0.5*dt*k2_omega2;

      let k3_theta1 = _omega1;
      let k3_theta2 = _omega2;
      let k3_omega1, k3_omega2;

      {
        let save_t1 = theta1, save_t2 = theta2, save_o1 = omega1, save_o2 = omega2;
        theta1 = _theta1; theta2 = _theta2; omega1 = _omega1; omega2 = _omega2;
        k3_omega1 = domega1();
        k3_omega2 = domega2();
        theta1 = save_t1; theta2 = save_t2; omega1 = save_o1; omega2 = save_o2;
      }

      _theta1 = theta1 + dt*k3_theta1;
      _theta2 = theta2 + dt*k3_theta2;
      _omega1 = omega1 + dt*k3_omega1;
      _omega2 = omega2 + dt*k3_omega2;

      let k4_theta1 = _omega1;
      let k4_theta2 = _omega2;
      let k4_omega1, k4_omega2;

      {
        let save_t1 = theta1, save_t2 = theta2, save_o1 = omega1, save_o2 = omega2;
        theta1 = _theta1; theta2 = _theta2; omega1 = _omega1; omega2 = _omega2;
        k4_omega1 = domega1();
        k4_omega2 = domega2();
        theta1 = save_t1; theta2 = save_t2; omega1 = save_o1; omega2 = save_o2;
      }

      theta1 += 1/daempf.Value()*dt*(k1_theta1 + 2*k2_theta1 + 2*k3_theta1 + k4_theta1);
      theta2 += 1/daempf.Value()*dt*(k1_theta2 + 2*k2_theta2 + 2*k3_theta2 + k4_theta2);
      omega1 += 1/daempf.Value()*dt*(k1_omega1 + 2*k2_omega1 + 2*k3_omega1 + k4_omega1);
      omega2 += 1/daempf.Value()*dt*(k1_omega2 + 2*k2_omega2 + 2*k3_omega2 + k4_omega2);

      board.update();
    }
    requestAnimationFrame(step);
  }
  step();

  // Rechteck als Button
  const button = board.create('polygon', [[-3.5,3.5],[-2.5,3.5],[-2.5,2.8],[-3.5,2.8]], {
    borders: {strokeWidth:1, strokeColor:'black'},
    fillColor:'lightgray',
    fillOpacity:0.6,
    withLines:true,
    hasInnerPoints:true
  });

  const label = board.create('text', [-3.4,3.3, () => running ? 'Stop' : 'Start'], {
    fontSize:16, anchorX:'left', anchorY:'top'
  });

  button.on('down', () => {
    board.update();
    running = !running;
  });
</jsxgraph>