\u3053\u306e\u8a18\u4e8b<\/a>\u3092\u30ac\u30a4\u30c9\u306b\u6570\u5b66\u3092\u9032\u3081\u308b\u3053\u3068\u306b\u3057\u305f\u3002<\/p>\n\u30ac\u30a4\u30c9\u8a18\u4e8b\u3092\u898b\u3066\u307f\u308b\u3068\u524d\u7ae0\u306e\u5f0f\uff080.1\uff09\u3068\uff080.2\uff09\u304c\u30ac\u30a4\u30c9\u8a18\u4e8b\u306e\u5f0f\uff0814\uff09\u3068\uff0819\uff09\u306b\u8a72\u5f53\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u30ac\u30a4\u30c9\u8a18\u4e8b\u306b\u306f\u5c0e\u51fa\u65b9\u6cd5\u306e\u6982\u8981\u304c\u66f8\u304b\u308c\u3066\u3044\u308b\u304c\u3001\u7d30\u304b\u3044\u3068\u3053\u308d\u306f\u7701\u304b\u308c\u3066\u3044\u308b\u304b\u3089\u30ac\u30a4\u30c9\u8a18\u4e8b\u306e\u8b70\u8ad6\u3092\u8ffd\u3044\u306a\u304c\u3089\u3067\u304d\u308b\u3060\u3051\u5206\u304b\u308a\u3084\u3059\u304f\u3001\u5f0f\u5909\u5f62\u306e\u9014\u4e2d\u7d50\u679c\u3082\u8f09\u305b\u3066\u8aac\u660e\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n
\u307e\u305a\u6700\u521d\u306b\u632f\u308a\u5b50\u306e\u305d\u308c\u305e\u308c\u306e\u8cea\u70b9 $m_1$ \u3068 $m_2$ \u306e\u4f4d\u7f6e\u3092xy\u5ea7\u6a19\u3067\u8868\u3059\u3068<\/p>\n
$$
\n\\begin{align}
\nx_1 &= l_1 \\sin{\\theta_1} \\\\
\ny_1 &= - l_1 \\cos{\\theta_1} \\\\
\nx_2 &= l_1 \\sin{\\theta_1} + l_2 \\sin{\\theta_2} \\\\
\ny_2 &= - l_1 \\cos{\\theta_1} - l_2 \\cos{\\theta_2}
\n\\tag{2.1}
\n\\end{align}
\n$$
\n\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u7d50\u679c\u306f Fig. 1 \u306b\u3057\u305f\u306e\u56f3\u306b\u3042\u308b\u3088\u3046\u306b\u76f4\u89d2\u4e09\u89d2\u5f62\u3092\u63cf\u3044\u3066\u4e09\u89d2\u95a2\u6570\uff08sin \u3068 cos\uff09\u306e\u5b9a\u7fa9\u3092\u9069\u7528\u3059\u308b\u3068\u6c42\u307e\u308b\u3002<\/p>\n
![\"\u56f32\"](\"https:\/\/virtualcast.jp\/blog\/wp-content\/uploads\/2020\/02\/double-pendulum-fig-2.png\")
<\/p>\n
\u6b21\u306f\u30b7\u30b9\u30c6\u30e0\u306e\u4f4d\u7f6e\u30a8\u30cd\u30eb\u30ae\u30fc $V$ \u3068\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc $T$ \u3092\u6c42\u3081\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308b\u3002\u5f8c\u307b\u3069\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa\u306b\u4f7f\u3044\u307e\u3059\u3002<\/p>\n
\u30b7\u30b9\u30c6\u30e0\u306e\u4f4d\u7f6e\u30a8\u30cd\u30eb\u30ae\u30fc $V$ \u306f<\/p>\n
$$
\nV = m_1 g y_1 + m_2 g y_2
\n\\tag{2.2}
\n$$<\/p>\n
\u3067\u3042\u308b\u3002(2.2) \u306e $y_1$ \u3068 $y_2$ \u306f (2.1) \u306e\u65b9\u7a0b\u5f0f\u306b\u3082\u51fa\u3066\u3044\u307e\u3059\u306d\u3002(2.1) \u306e\u4ee5\u4e0b\u306e\u4e8c\u3064\u306e\u5f0f\u3092<\/p>\n
$$
\n\\begin{align}
\ny_1 &= - l_1 \\cos{\\theta_1} \\\\
\ny_2 &= - l_1 \\cos{\\theta_1} - l_2 \\cos{\\theta_2}
\n\\end{align}
\n$$<\/p>\n
(2.2) \u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n
$$
\n\\begin{align}
\nV &= m_1 g (-l_1 \\cos{\\theta_1}) + m_2 g (-l_1 \\cos{\\theta_1} - l_2 \\cos{\\theta_2})
\n\\\\
\n&= -m_1 g l_1 \\cos{\\theta_1} - m_2 g l_1 \\cos{\\theta_1} - m_2 g l_2 \\cos{\\theta_2})
\n\\\\
\n&= -(m_1 + m_2)g l_1 \\cos(\\theta_1) - m_2 g l_2 \\cos{\\theta_2}
\n\\tag{2.3}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u6b21\u306f\u30b7\u30b9\u30c6\u30e0\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc $T$ \u3067\u3059\u3002\u53e4\u5178\u529b\u5b66\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5b9a\u7fa9\u304c
\n$$
\nE_k = \\frac{1}{2}mv^2
\n$$
\n\u3067\u3042\u308b\u3053\u3068\u3092\u601d\u3044\u51fa\u3057\u3066\u304a\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u4e00\u5ea6\u79c1\u305f\u3061\u304c\u8003\u5bdf\u3057\u3066\u3044\u308b\u30b7\u30b9\u30c6\u30e0\u306e\u8cea\u70b9 $m_1$ \u3068 $m_2$ \u306e\u305d\u308c\u305e\u308c\u306e\u901f\u5ea6 $v_1$ \u3068 $v_2$ \u304c\u3069\u306e\u3088\u3046\u306b\u6c42\u307e\u308b\u304b\u306b\u3064\u3044\u3066\u8003\u3048\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u89d2\u5ea6 $\\theta$ \u306e\u89d2\u901f\u5ea6\u304c $\\dot{\\theta}$ \u306b\u306a\u308a\u307e\u3059\u3002\u305d\u3057\u3066\u539f\u70b9\u304b\u3089\u8ddd\u96e2 $r$ \u96e2\u308c\u305f\u8cea\u70b9\u3092\u89d2\u901f\u5ea6 $\\dot{\\theta}$ \u3067\u539f\u70b9\u3092\u4e2d\u5fc3\u306b\u56de\u3059\u3068\u8cea\u70b9\u306e\u901f\u5ea6 $v$ \u304c
\n$$
\nv = \\dot{\\theta} r
\n\\tag{2.4}
\n$$
\n\u306b\u306a\u308b\u306e\u3067
\n$$
\nv_1 = l_1 \\dot{\\theta}_1
\n\\tag{2.5}
\n$$
\n\u3068 \u8cea\u70b9 $m_1$ \u306e\u901f\u5ea6 $v_1$ \u304c\u6c42\u307e\u308a\u307e\u3057\u305f\u3002\u6b21\u306b $v_2$ \u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002(2.1) \u306e\u4ee5\u4e0b\u306e\u4e8c\u3064\u306e\u65b9\u7a0b\u5f0f\u304b\u3089\u30b9\u30bf\u30fc\u30c8\u3057\u307e\u3059\u3002
\n$$
\n\\begin{align}
\nx_2 &= l_1 \\sin{\\theta_1} + l_2 \\sin{\\theta_2} \\\\
\ny_2 &= - l_1 \\cos{\\theta_1} - l_2 \\cos{\\theta_2}
\n\\tag{2.6}
\n\\end{align}
\n$$<\/p>\n
$x_2$ \u3068 $y_2$ \u3092\u6642\u9593 $t$ \u3067\u5fae\u5206\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u305d\u308c\u305d\u308c\u306e\u6210\u5206\u306e\u901f\u5ea6 $v_{x_2}$ \u3068 $v_{y_2}$\u00a0\u3092\u6c42\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n
$$
\n\\begin{align}
\nv_{x_2} =\u00a0l_1 \\dot{\\theta}_1\u00a0 \\cos{\\theta_1} + l_2 \\dot{\\theta}_2 \\cos{\\theta_2}\\\\
\nv_{y_2} =\u00a0l_1 \\dot{\\theta}_1 \\sin{\\theta_1} + l_2 \\dot{\\theta}_2 \\sin{\\theta_2}
\n\\tag{2.7}
\n\\end{align}
\n$$<\/p>\n
\u3053\u3053\u3067\u300c$t$ \u3067\u5fae\u5206\u3059\u308b\u3063\u3066\u8a00\u3063\u3066\u3082\u3001(2.6) \u306b $t$ \u306a\u3093\u304b\u3069\u3053\u306b\u3082\u306a\u3044\u305e\uff01\u300d\u3068\u601d\u3046\u4eba\u304c\u3044\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u3053\u308c\u306f $\\theta(t)$ \u3092 $\\theta$ \u3068\u7701\u7565\u3057\u3066\u308b\u304b\u3089\u3067\u3059\u3002\u3053\u306e\u8a18\u4e8b\u306e $\\theta$ \u306f\u307f\u3093\u306a\u672c\u5f53\u306f $\\theta(t)$ \u3068\u601d\u3063\u3066\u304f\u3060\u3055\u3044\u3002\u6570\u5f0f\u304c\u898b\u3084\u3059\u304f\u306a\u308b\u304b\u3089\u7701\u7565\u3057\u3066\u3044\u308b\u308f\u3051\u3067\u3059\u3002<\/p>\n
(2.6) \u3092\u6642\u9593 $t$ \u3067\u5fae\u5206\u3059\u308b\u3068\u304d\u9023\u9396\u5f8b
\n$$
\n\\frac{d}{dt}f(g(t)) = f'(g(t)) g'(t)
\n$$
\n\u304c\u9069\u7528\u3055\u308c\u308b\u3002<\/p><\/blockquote>\n
\u3055\u3066\u00a0(2.7) \u306e\u305d\u308c\u305e\u308c\u306e\u901f\u5ea6\u6210\u5206\u304b\u3089 $v_2$ \u3092\u6c42\u3081\u3088\u3046\u3002\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u304b\u3089
\n$$
\nv_2 = \\sqrt{v_{x_2}^2 + v<\/em>{y_2}^2}
\n\\tag{2.8}
\n$$<\/p>\n
\u305d\u3057\u3066 (2.8) \u306b (2.7) \u3092\u4ee3\u5165\u3057\u3066\u6574\u7406\u3059\u308b\u3068<\/p>\n
$$
\n\\begin{align}
\nv_2 &= \\sqrt{(l_1 \\dot{\\theta}_1 \\cos{\\theta_1} + l_2 \\dot{\\theta}_2 \\cos{\\theta_2})^2(l_1 \\dot{\\theta}_1 \\sin{\\theta_1} + l_2 \\dot{\\theta}_2 \\sin{\\theta_2})^2} \\\\
\n&= \\sqrt{l_1^2 \\dot{\\theta}^2_1 (\\cos^2{\\theta_1} + \\sin^2{\\theta_1}) + l_2^2 \\dot{\\theta}^2_2\u00a0(\\cos^2{\\theta_2} + \\sin^2{\\theta_2}) + l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2)} \\\\
\n&= \\sqrt{l_1^2 \\dot{\\theta}^2_1 + l_2^2 \\dot{\\theta}^2_2 + l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2)}
\n\\tag{2.9}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u308b\u3002\u3053\u308c\u3067 $v_1$ \u3068 $v_2$ \u304c\u6c42\u307e\u3063\u305f\u306e\u3067\u3084\u3063\u3068\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc $T$ \u306b\u623b\u308c\u307e\u3059\u3002\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5b9a\u7fa9
\n$$
\nE_k = \\frac{1}{2}mv^2
\n$$
\n\u304b\u3089\u79c1\u305f\u3061\u304c\u8003\u5bdf\u3057\u3066\u3044\u308b\u30b7\u30b9\u30c6\u30e0\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc $T$ \u306f
\n$$
\nT = \\frac{1}{2}m_1 v_1^2 +\u00a0\\frac{1}{2}m_2 v_2^2
\n\\tag{2.10}
\n$$
\n\u3068\u6c42\u307e\u308a\u307e\u3059\u3002<\/p>\n
(2.10) \u306b (2.5) \u3067\u6c42\u307e\u3063\u305f $v_1$ \u3068 (2.9) \u3067\u6c42\u307e\u3063\u305f $v_2$ \u3092\u4ee3\u5165\u3059\u308b\u3068
\n$$
\nT = \\frac{1}{2}m_1 l_1^2 \\dot{\\theta}_1^2 +\u00a0\\frac{1}{2}m_2 \\left(l_1^2 \\dot{\\theta}^2_1 + l_2^2 \\dot{\\theta}^2_2 + 2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2)\\right)
\n\\tag{2.11}
\n$$
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
\u3053\u308c\u3067\u30b7\u30b9\u30c6\u30e0\u306e\u4f4d\u7f6e\u30a8\u30cd\u30eb\u30ae\u30fc $V$ \u3068\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc $T$ \u304c\u6c42\u307e\u3063\u305f\uff01\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u5c0e\u51fa\u3059\u308b\u305f\u3081\u306b Lagrangian Mechanics \u306e\u6570\u5b66\u3092\u4f7f\u3044\u307e\u3059\u3002\u4e00\u822c\u7684\u306b Newtonian Mechanics \u3092\u7528\u3044\u3066\u529b\u306e\u91e3\u308a\u5408\u3044\u3068 $F = ma$ \u306e\u95a2\u4fc2\u304b\u3089\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u5c0e\u51fa\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\u3001\u4e8c\u91cd\u632f\u308a\u5b50\u306e\u3088\u3046\u306b\u3001\u8cea\u70b9\u306e\u8ecc\u9053\u304c\u5236\u7d04\u3055\u308c\u3066\u3044\u308b\u30b7\u30b9\u30c6\u30e0\u306e\u5834\u5408\u3001 Lagrangian Mechanics \u3092\u4f7f\u3063\u305f\u65b9\u304c\u90fd\u5408\u304c\u3044\u3044\u3002\u3068\u3044\u3046\u3053\u3068\u3067 Lagrangian Mechanics \u306e\u4e16\u754c\u306b\u7a81\u5165\uff01<\/p>\n
Lagrangian $\\mathcal{L}$ \u304c
\n$$
\n\\mathcal{L} =\u00a0 T - V
\n\\tag{2.12}
\n$$
\n\u3068\u5b9a\u7fa9\u3055\u308c\u308b\u3002<\/p>\n
(2.12) \u306b (2.3) \u3068 (2.11) \u3092\u4ee3\u5165\u3059\u308b\u3068
\n$$
\n\\begin{align}
\n\\mathcal{L} =&\u00a0\u00a0\\frac{1}{2}m_1 l_1^2 \\dot{\\theta}_1^2 +\u00a0\\frac{1}{2}m_2 \\left(l_1^2 \\dot{\\theta}^2_1 + l_2^2 \\dot{\\theta}^2_2 + 2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2)\\right)\u00a0 \\\\
\n&- \\left(\u00a0-(m_1 + m_2)g l_1 \\cos(\\theta_1) - m_2 g l_2 \\cos{\\theta_2} \\right) \\\\
\n=& \\frac{1}{2}(m_1 + m_2) l_1^2 \\dot{\\theta}^2_1 + \\frac{1}{2}m_2l_2^2\\dot{\\theta}^2_2 + m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) \\\\
\n&+(m_1 + m_2)g l_1 cos{\\theta_1} + m_2 g l_2 \\cos{\\theta_2}
\n\\tag{2.13}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u6b21\u306f Euler-Lagrange\u00a0\u65b9\u7a0b\u5f0f\u306e\u767b\u5834\u3067\u3059\u3002\u3053\u306e\u65b9\u7a0b\u5f0f\u306f Newtonian Mechanics \u306e\u6709\u540d\u306a\u904b\u52d5\u306e\u6cd5\u5247\u306e Lagrangian Mechanics \u7248\u3067\u3059\u3002$\\theta_1$ \u306b\u3064\u3044\u3066\u306e Euler-Lagrange\u00a0\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$
\n\\frac{d}{dt}\\left(\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_1}\\right) -\u00a0\\frac{\\partial \\mathcal{L}}{\\partial \\theta_1} = 0
\n\\tag{2.14}
\n$$<\/p>\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
\u3082\u3046\u5c11\u3057\u3067\u4e8c\u91cd\u632f\u308a\u5b50\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u6c42\u307e\u308a\u307e\u3059\uff01<\/p>\n
(2.14) \u306e\u8a08\u7b97\u3092\u4e09\u3064\u306b\u5206\u3051\u308b\u3068\u3084\u308a\u3084\u3059\u3044\u306e\u3067<\/p>\n
$$
\n\\begin{align}
\n&\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_1} \\tag{2.15}\\\\
\n&\\frac{d}{dt}\\left(\u00a0\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_1} \\right)\u00a0 \\tag{2.16}\\\\
\n&\\frac{\\partial \\mathcal{L}}{\\partial \\theta_1} \\tag{2.17}\\\\
\n\\end{align}
\n$$
\n\u3068\u8a08\u7b97\u91cf\u3092\u5206\u5272\u3057\u3088\u3046\u3002<\/p>\n
\u307e\u305a (2.15) \u306b (2.13) \u3092\u4ee3\u5165\u3057\u3066\u504f\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u3068
\n$$
\n\\begin{align}
\n\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_1} =&
\n\\frac{\\partial}{\\partial \\dot{\\theta}_1} \\bigg(\\frac{1}{2}(m_1 + m_2) l_1^2 \\dot{\\theta}^2_1 + \\frac{1}{2}m_2l_2^2\\dot{\\theta}^2_2 + m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) \\\\
\n&+(m_1 + m_2)g l_1 cos{\\theta_1} + m_2 g l_2 \\cos{\\theta_2}\\bigg) \\\\
\n=& (m_1 + m_2)l_1^2 \\dot{\\theta}_1 + m_2 l_1 l_2 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) \\tag{2.18}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u6b21\u306b (2.16) \u306b (2.18) \u3092\u4ee3\u5165\u3057\u3066\u5fae\u5206\u3092\u5b9f\u884c\u3059\u308b\u3068
\n$$
\n\\begin{align}
\n\\frac{d}{dt}\\left(\u00a0\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_1} \\right)\u00a0 =& \\frac{d}{dt}\\left(\u00a0(m_1 + m_2)l_1^2 \\dot{\\theta}_1 + m_2 l_1 l_2 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) \u00a0\\right) \\tag{2.19}\\\\
\n=& (m_1 + m_2) l_1^2 \\ddot{\\theta}_1 + m_2 l_1 l_2 \\ddot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) \\\\
\n&- m_2 l_1 l_2 \\dot{\\theta}_2 \\sin(\\theta_1 - \\theta_2) (\\dot{\\theta}_1 - \\dot{\\theta}_2)\u00a0\\tag{2.20}
\n\\end{align}
\n$$
\n\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
(2.19) \u306e\u5fae\u5206\u3092\u5b9f\u884c\u3059\u308b\u3068\u304d\u306b\u9023\u9396\u5f8b
\n$$
\n\\frac{d}{dt}f(g(t)) = f'(g(t)) g'(t)
\n$$
\n\u3068\u7a4d\u306e\u6cd5\u5247
\n$$
\n\\frac{d}{dt}f(t)g(t) = f'(t)g(t) + f(t)g'(t)
\n$$
\n\u304c\u9069\u7528\u3055\u308c\u308b\u3002<\/p><\/blockquote>\n
\u6700\u5f8c\u306b (2.17) \u306b (2.13) \u3092\u4ee3\u5165\u3057\u3066\u504f\u5fae\u5206\u3092\u5b9f\u884c\u3059\u308b\u3068
\n$$
\n\\begin{align}
\n\\frac{\\partial \\mathcal{L}}{\\partial \\theta_1} =&\u00a0 \\frac{\\partial}{\\partial \\theta_1}
\n\\bigg(\\frac{1}{2}(m_1 + m_2) l_1^2 \\dot{\\theta}^2_1 + \\frac{1}{2}m_2l_2^2\\dot{\\theta}^2_2 + m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) \\\\
\n&+(m_1 + m_2)g l_1 cos{\\theta_1} + m_2 g l_2 \\cos{\\theta_2}\\bigg) \\\\
\n=& -m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\sin(\\theta_1 - \\theta_2) - (m_1 + m_2) g l_1 \\sin{\\theta_1} \\tag{2.21}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u3053\u308c\u3067 (2.14) \u306e\u6e96\u5099\u306e\u305f\u3081\u306e\u8a08\u7b97\u304c\u7d42\u308f\u308a\u307e\u3057\u305f\u3002 (2.14) \u3092\u3082\u3046\u4e00\u5ea6\u66f8\u3044\u3066\u304a\u304d\u307e\u3059\u3002
\n$$
\n\\frac{d}{dt}\\left(\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_1}\\right) -\u00a0\\frac{\\partial \\mathcal{L}}{\\partial \\theta_1} = 0
\n\\tag{2.14}
\n$$<\/p>\n
(2.14) \u306b (2.20) \u3068 (2.21) \u3092\u4ee3\u5165\u3057\u3066\u6574\u7406\u3059\u308b\u3068<\/p>\n
$$
\n\\begin{align}
\n&(m_1 + m_2) l_1^2 \\ddot{\\theta}_1 + m_2 l_1 l_2 \\ddot{\\theta}_2 \\cos(\\theta_1 - \\theta_2)m_2 l_1 l_2 \\dot{\\theta}_2 \\sin(\\theta_1 - \\theta_2) (\\dot{\\theta}_1 - \\dot{\\theta}_2) \\\\
\n&- \\left(-m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\sin(\\theta_1 - \\theta_2) - (m_1 + m_2) g l_1 \\sin{\\theta_1}\\right) \\\\
\n\\\\
\n=& (m_1 + m_2) l_1^2 \\ddot{\\theta}_1 + m_2 l_1 l_2 \\ddot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) - m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\sin(\\theta_1 - \\theta_2) \\\\
\n& m_2 l_1 l_2 \\dot{\\theta}^2_2\\sin(\\theta_1 - \\theta_2) + m_2 l_1 l_2 \\dot{\\theta}_1 \\dot{\\theta}_2 \\sin(\\theta_1 - \\theta_2) + (m_1 + m_2) g l_1 \\sin{\\theta_1} \\\\
\n\\\\
\n=& (m_1 + m_2) l_1^2 \\ddot{\\theta}_1 + m_2 l_1 l_2 \\ddot{\\theta}_2 \\cos(\\theta_1 - \\theta_2) + m_2 l_1 l_2 \\dot{\\theta}^2_2 \\sin(\\theta_1 - \\theta_2) \\\\
\n&+ (m_1 + m_2) g l_1 \\sin{\\theta_1} = 0 \\tag{2.22}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u6700\u5f8c\u306b (2.22) \u306e\u4e21\u8fba\u3092 $l_1$ \u3067\u5272\u308b\u3068
\n$$
\n\\begin{align}
\n(m_1 + m_2) l_1 \\ddot{\\theta}_1 + m_2 l_2 \\ddot{\\theta}_2 \\cos(\\theta_1 - \\theta_2)
\n\\\\+m_2 l_2 \\dot{\\theta}^2_2 \\sin(\\theta_1 - \\theta_2) + g (m_1 + m_2) \\sin \\theta_1 &= 0
\n\\tag{0.1}
\n\\end{align}
\n$$
\n\u3068\u3001$\\theta_1$ \u306b\u3064\u3044\u3066\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\uff01<\/p>\n
\u540c\u69d8\u306b\u3001$\\theta_2$ \u306b\u304a\u3051\u308b Euler-Lagrange\u00a0\u65b9\u7a0b\u5f0f
\n$$
\n\\frac{d}{dt}\\left(\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}_2}\\right) -\u00a0\\frac{\\partial \\mathcal{L}}{\\partial \\theta_2} = 0
\n\\tag{2.23}
\n$$
\n\u306e\u5fae\u5206\u3068\u504f\u5fae\u5206\u3092\u5b9f\u884c\u3059\u308b\u3068
\n$$
\n\\begin{align}
\nm_2 l_2 \\ddot{\\theta}_2 + m_2 l_1 \\ddot{\\theta}_1 cos(\\theta_1 - \\theta_2) \\\\
\n-m_2 l_1 \\dot{\\theta}^2_1 sin(\\theta_1 - \\theta_2) + m_2 g \\sin \\theta_2 &= 0
\n\\tag{0.2}
\n\\end{align}
\n$$
\n\u3068 $\\theta_2$ \u306b\u3064\u3044\u3066\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u6c42\u307e\u308a\u307e\u3059\u3002<\/p>\n
3. Euler \u6cd5\u3092\u4f7f\u3046\u305f\u3081\u306e\u6e96\u5099<\/h2>\n
Euler \u6cd5\u3067\u524d\u7ae0\u3067\u5f97\u3089\u308c\u305f\u4e8c\u91cd\u632f\u308a\u5b50\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3092\u884c\u3044\u305f\u3044\u304c\u3001\u5b9f\u306f (0.1) \u3068 (0.2) \u3092\u305d\u306e\u307e\u307e\u4f7f\u3048\u306a\u3044\u306e\u3067\u3059\u3002Euler \u6cd5\u3067\u4e00\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u8a08\u7b97\u7684\u306b\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3067\u304d\u308b\u304c\u3001(0.1) \u3068 (0.2) \u306f\u3069\u3061\u3089\u3082\u4e8c\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3002<\/p>\n
Euler \u6cd5\u304c\u4f7f\u3048\u308b\u3088\u3046\u306b (0.1) \u3068 (0.2) \u305d\u308c\u305e\u308c\u306e\u4e8c\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u4e8c\u3064\u306e\u4e00\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5206\u3051\u308c\u3070\u3044\u3044\u3002\u3069\u3046\u3059\u308c\u3070\u5206\u5272\u3067\u304d\u308b\uff1f\u3068\u7591\u554f\u306b\u601d\u3046\u4eba\u3082\u3044\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002\u65b0\u3057\u3044\u5909\u6570\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308c\u3070\u3088\u3044\u3002
\n$$
\n\\begin{align}
\n\\lambda_1\u00a0&=\u00a0\\dot{\\theta}_1\u00a0\\tag{3.1}\u00a0\\\\
\n\\dot{\\lambda}_1\u00a0&=\u00a0\\ddot{\\theta}_1 \\tag{3.2}
\n\\\\
\n\\lambda_2\u00a0&=\u00a0\\dot{\\theta}_2 \\tag{3.3} \\\\
\n\\dot{\\lambda}_2\u00a0&=\u00a0\\ddot{\\theta}_2 \\tag{3.4}
\n\\end{align}
\n$$<\/p>\n
(3.1) ~ (3.4) \u3092 (0.1)\u00a0\u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n
$$
\n\\begin{align}
\n(m_1\u00a0+\u00a0m_2)\u00a0l_1\u00a0\\dot{\\lambda}_1\u00a0+\u00a0m_2\u00a0l_2\u00a0\\dot{\\lambda}_2\u00a0\\cos(\\theta_1\u00a0-\u00a0\\theta_2) \\\\
\n+\u00a0m_2\u00a0l_2\u00a0\\lambda^2_2\u00a0\\sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0+\u00a0g\u00a0(m_1\u00a0+\u00a0m_2)\u00a0\\sin\u00a0\\theta_1 &=\u00a00
\n\\tag{3.5}
\n\\end{align}
\n$$<\/p>\n
\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u6b21\u306b (3.5) \u3092 $\\dot{\\lambda}_1$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n
$$
\n\\begin{align}
\n\\dot{\\lambda}_1\u00a0&=\u00a0\\frac{-\u00a0m_2\u00a0l_2\u00a0\\dot{\\lambda}_2\u00a0\\cos(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0l_2\u00a0\\lambda^2_2\u00a0\\sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0g\u00a0(m_1\u00a0+\u00a0m_2)\u00a0\\sin\u00a0\\theta_1}{(m_1\u00a0+\u00a0m_2)\u00a0l_1}
\n\\tag{3.6}
\n\\end{align}
\n$$<\/p>\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
\u540c\u69d8\u306b (3.1) ~ (3.4) \u3092 (0.2)\u00a0\u306b\u4ee3\u5165\u3057\u3066<\/p>\n
$$
\n\\begin{align}
\nm_2\u00a0l_2\u00a0\\dot{\\lambda}_2\u00a0+\u00a0m_2\u00a0l_1\u00a0\\dot{\\lambda}_1\u00a0cos(\\theta_1\u00a0-\u00a0\\theta_2)
\n-\u00a0m_2\u00a0l_1\u00a0\\lambda^2_1\u00a0sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0+\u00a0m_2\u00a0g\u00a0\\sin\u00a0\\theta_2\u00a0&=\u00a00
\n\\tag{3.7}
\n\\end{align}
\n$$<\/p>\n
(3.7) \u3092 $\\dot{\\lambda}_2$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n
$$
\n\\begin{align}
\n\\dot{\\lambda}_2\u00a0&=\u00a0\\frac{m_2\u00a0l_1\u00a0\\lambda^2_1\u00a0sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0l_1\u00a0\\dot{\\lambda}_1\u00a0cos(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0g\u00a0\\sin\u00a0\\theta_2}{m_2\u00a0l_2}
\n\\tag{3.8}
\n\\end{align}
\n$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u3053\u308c\u3067\u305d\u308c\u305e\u308c\u306e\u4e8c\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u305d\u308c\u305e\u308c\u4e8c\u3064\u306e\u4e00\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5206\u5272\u3059\u308b\u3053\u3068\u306b\u6210\u529f\u3057\u307e\u3057\u305f\u3002\u5206\u5272\u5f8c\u306e\u56db\u3064\u306e\u4e00\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u3082\u3046\u4e00\u5ea6\u307e\u3068\u3081\u3066\u66f8\u3044\u3066\u304a\u304f\u3068
\n$$
\n\\begin{align}
\n\\dot{\\theta}_1\u00a0 &=\u00a0\\lambda_1 \\tag{3.1}\u00a0\\\\
\n\\dot{\\lambda}_1\u00a0&=\u00a0\\frac{-\u00a0m_2\u00a0l_2\u00a0\\dot{\\lambda}_2\u00a0\\cos(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0l_2\u00a0\\lambda^2_2\u00a0\\sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0g\u00a0(m_1\u00a0+\u00a0m_2)\u00a0\\sin\u00a0\\theta_1}{(m_1\u00a0+\u00a0m_2)\u00a0l_1}
\n\\tag{3.6} \\\\
\n\\dot{\\theta}_2 &=\u00a0\\lambda_2 \\tag{3.3} \\\\
\n\\dot{\\lambda}_2\u00a0&=\u00a0\\frac{m_2\u00a0l_1\u00a0\\lambda^2_1\u00a0sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0l_1\u00a0\\dot{\\lambda}_1\u00a0cos(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0g\u00a0\\sin\u00a0\\theta_2}{m_2\u00a0l_2}
\n\\tag{3.8} \\\\
\n\\end{align}
\n$$
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
\u3053\u306e\u56db\u3064\u306e\u65b9\u7a0b\u5f0f\u306b Newton \u8a18\u6cd5\u306e\u5fae\u5206\u70b9\u304c\u4e8c\u3064\u4ed8\u3044\u3066\u3044\u308b\u5909\u6570\u304c\u4e00\u3064\u3082\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u307b\u3057\u3044\u3002\u307f\u3093\u306a\u304c\u4e00\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\u3053\u308c\u3089\u306b Euler \u6cd5\u3092\u9069\u7528\u3067\u304d\u307e\u3059\uff01<\/p>\n
4. Euler \u6cd5\u3067\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3059\u308b<\/h2>\n
\u3084\u3063\u3068 Euler \u6cd5\u3092\u9069\u7528\u3059\u308b\u3068\u3053\u308d\u307e\u3067\u6765\u307e\u3057\u305f\u3002\u307e\u305a Euler \u6cd5\u306e\u4ed5\u7d44\u307f\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\u3002
\n$$
\n\\begin{align}
\n\\dot{y}\u00a0&=\u00a0f(y,\u00a0t)\u00a0\\tag{4.1}\\\\
\ny(t_0)\u00a0&=\u00a0y_0 \\tag{4.2}
\n\\end{align}
\n$$
\n(4.1) \u3067\u8868\u305b\u308b\u3088\u3046\u306a\u4e00\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u3042\u308b\u3068\u3057\u3088\u3046\u3002\u305d\u3057\u3066 (4.2) \u3067\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u72b6\u614b\u3092\u5b9a\u7fa9\u3057\u3088\u3046\u3002\u3053\u306e\u3068\u304d\u306b\u3001Euler \u6cd5\u3067\u6642\u9593\u3092 $\\epsilon$ \u3060\u3051\u9032\u3081\u308b\u3068
\n$$
\n\\begin{align}
\nt_1\u00a0&=\u00a0t_0\u00a0+\u00a0\\epsilon\u00a0\\\\
\ny_1\u00a0&=\u00a0y_0\u00a0+\u00a0\\epsilon\u00a0f(y_0,\u00a0t_0) \\tag{4.3}
\n\\end{align}
\n$$
\n$y_1$ \u304c\u5b9a\u307e\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $\\epsilon$ \u306f\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u306e\u30bf\u30a4\u30e0\u30b9\u30c6\u30c3\u30d7\u3067\u3001\u5c0f\u3055\u3051\u308c\u3070\u5c0f\u3055\u3044\u307b\u3069\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u304c\u3088\u308a\u6b63\u78ba\u306b\u306a\u308b\u3002\u4eca\u56de\u306e\u4e8c\u91cd\u632f\u308a\u5b50\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3067\u306f\u300c0.0001\u300d\u3068\u8a2d\u5b9a\u3057\u307e\u3057\u305f\u3002(4.3) \u306f\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3092\u958b\u59cb\u3055\u305b\u3066\u4e00\u756a\u6700\u521d\u306e\u30b9\u30c6\u30c3\u30d7\u3092\u5b9f\u884c\u3055\u305b\u305f\u3068\u304d\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u305d\u306e\u5f8c\u3082\u540c\u3058\u30d1\u30bf\u30fc\u30f3\u3067\u9032\u307f\u307e\u3059\u3002<\/p>\n
\u3053\u3053\u3067\u91cd\u8981\u306a\u306e\u304c\u3001Euler \u6cd5\u304c $\\dot{y}_n =\u00a0f(y_n,\u00a0t_n)$ \u3092\u5165\u529b\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3066 $y_{n+1}$ \u3092\u8fd4\u3057\u3066\u304f\u308c\u308b\u3068\u3053\u308d\u3067\u3059\uff01\u3064\u307e\u308a\u3001 $\\dot{y}$ \u304b\u3089 $y$ \u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n
\u3082\u3046\u4e00\u5ea6\u6e96\u5099\u3057\u305f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n
$$
\n\\begin{align}
\n\\dot{\\theta}_1\u00a0 &=\u00a0\\lambda_1 \\tag{3.1}\u00a0\\\\
\n\\dot{\\lambda}_1\u00a0&=\u00a0\\frac{-\u00a0m_2\u00a0l_2\u00a0\\dot{\\lambda}_2\u00a0\\cos(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0l_2\u00a0\\lambda^2_2\u00a0\\sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0g\u00a0(m_1\u00a0+\u00a0m_2)\u00a0\\sin\u00a0\\theta_1}{(m_1\u00a0+\u00a0m_2)\u00a0l_1}
\n\\tag{3.6} \\\\
\n\\dot{\\theta}_2 &=\u00a0\\lambda_2 \\tag{3.3} \\\\
\n\\dot{\\lambda}_2\u00a0&=\u00a0\\frac{m_2\u00a0l_1\u00a0\\lambda^2_1\u00a0sin(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0l_1\u00a0\\dot{\\lambda}_1\u00a0cos(\\theta_1\u00a0-\u00a0\\theta_2)\u00a0-\u00a0m_2\u00a0g\u00a0\\sin\u00a0\\theta_2}{m_2\u00a0l_2}
\n\\tag{3.8} \\\\
\n\\end{align}
\n$$<\/p>\n
$\\dot{\\lambda}_1$ \u3092 Euler \u6cd5\u3067\u51e6\u7406\u3057\u3066 $\\lambda_1$ \u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3055\u3089\u306b $\\dot{\\theta}_1\u00a0 =\u00a0\\lambda_1 $ \u306e\u95a2\u4fc2\u3092\u7528\u3044\u3066 Euler \u6cd5\u3067 $\\dot{\\theta}_1$ \u3092\u51e6\u7406\u3057\u3066 $\\theta_1$ \u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u540c\u69d8\u306b $\\theta_2$ \u3082\u5f97\u3089\u308c\u308b\u306e\u3067 $\\theta_1$ \u3068 $\\theta_2$ \u3092\u6bce\u30d5\u30ec\u30fc\u30e0 Unity \u306e\u4e2d\u3067\u4f5c\u3063\u3066\u304a\u3044\u305f\u4e8c\u91cd\u632f\u308a\u5b50\u306b\u9069\u7528\u3059\u308b\u3068\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u5b8c\u6210\u3067\u3059\uff01<\/p>\n