\u3068\u3044\u3046\u3002<\/p>\n
\u3067\u306f\uff0c\u632f\u5e45\u304c\u5927\u304d\u3044\u5834\u5408\uff08\u300c\u5341\u5206\u306b\u5c0f\u3055\u3044\u300d\u3068\u3044\u3046\u8fd1\u4f3c\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u5834\u5408\uff09\uff0c\u5358\u632f\u308a\u5b50\u306e\u5468\u671f\u306f\u632f\u5e45\u306b\u3069\u306e\u3088\u3046\u306b\u4f9d\u5b58\u3059\u308b\u304b\u3002<\/p>\n
\u554f\u984c\uff083\u629e\uff09<\/span><\/h3>\n\u307e\u305a\u306f\u4e88\u60f3\u3092\u7acb\u3066\u3066\u307f\u3088\u3046\u3002<\/p>\n
\u5358\u632f\u308a\u5b50\u306e\u632f\u5e45\u304c\u5927\u304d\u304f\u306a\u308b\u3068\uff08\u300c\u5341\u5206\u306b\u5c0f\u3055\u3044\u300d\u3068\u3044\u3046\u8fd1\u4f3c\u304c\u6210\u308a\u7acb\u305f\u306a\u304f\u306a\u308b\u3068\uff09\uff0c\u5358\u632f\u308a\u5b50\u306e\u5468\u671f\u306f\uff0c\u5358\u632f\u52d5\u306e\u5834\u5408\u306e\u5468\u671f\u306b\u304f\u3089\u3079\u3066…<\/p>\n
\n- \u300c\u632f\u308a\u5b50\u306e\u7b49\u6642\u6027\u300d\u304c\u3042\u308b\u304b\u3089\u5909\u5316\u3057\u306a\u3044\uff0c\u4e00\u5b9a\u306e\u307e\u307e\u3067\u3042\u308b\u3002<\/li>\n
- \u632f\u5e45\u304c\u5927\u304d\u304f\u306a\u308c\u3070\u306a\u308b\u307b\u3069\uff0c\u5468\u671f\u306f\u9577\u304f\u306a\u308b\u3002<\/li>\n
- \u632f\u5e45\u304c\u5927\u304d\u304f\u306a\u308c\u3070\u306a\u308b\u307b\u3069\uff0c\u5468\u671f\u306f\u77ed\u304f\u306a\u308b\u3002<\/li>\n<\/ol>\n
<\/p>\n
\u5358\u632f\u308a\u5b50\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa<\/h3>\n
\u5358\u3075\u308a\u3053\u306e\u3072\u3082\u306e\u9577\u3055\u3092 $\\ell$\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u306e\u5927\u304d\u3055\u3092 $g$\uff0c\u925b\u76f4\u4e0b\u5411\u304d\u304b\u3089\u306e\u632f\u308c\u89d2\u3092 $\\theta$ \u3068\u3059\u308b\u3068\uff0c\u5358\u3075\u308a\u3053\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$
\n\\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\sin\\theta
\n$$<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n
![](\"https:\/\/home.hirosaki-u.ac.jp\/jupyter\/pfuriko\/furiko.png\")
<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h4>\n
\u5358\u3075\u308a\u3053\u306e\u5148\u7aef\u306e\u304a\u3082\u308a\u306f\uff0c\u3072\u3082\u306e\u5f35\u529b $\\boldsymbol{T}$ \u3068\u91cd\u529b $ m\\boldsymbol{g}$ \u3092\u53d7\u3051\u308b\u3002\u925b\u76f4\u4e0b\u5411\u304d\u3092 $+z$ \u65b9\u5411\u3068\u3059\u308b\u3068 $\\boldsymbol{g} = (0, 0, g)$ \u306f\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb, $g$ \u306f\u91cd\u529b\u52a0\u901f\u5ea6\u306e\u5927\u304d\u3055\u3067\u3042\u308b\u3002<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$m \\frac{d^2\\boldsymbol{r}}{dt^2} = m\\boldsymbol{g} + \\boldsymbol{T}$$<\/p>\n
\u4fdd\u5b58\u91cf\uff08\u89d2\u904b\u52d5\u91cf\u306e\u925b\u76f4\u6210\u5206\uff09<\/h4>\n
\u3053\u306e\u5f0f\u306e\u4e21\u8fba\u306b\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306e\u5916\u7a4d\u3092\u304b\u3051\u308b\u3002$\\boldsymbol{T} \\propto -\\boldsymbol{r}$ \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068\uff0c$\\boldsymbol{r}\\times \\boldsymbol{T} = \\boldsymbol{0}$ \u3067\u3042\u308b\u304b\u3089\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
$$\\frac{d}{dt}\\left(m \\boldsymbol{r}\\times\\frac{d\\boldsymbol{r}}{dt}\\right) = m \\boldsymbol{r}\\times\\boldsymbol{g}$$<\/p>\n
\u3053\u306e\u4e21\u8fba\u306b\u4e00\u5b9a\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb $\\boldsymbol{g}$ \u306e\u5185\u7a4d\u3092\u304b\u3051\u3066\u3084\u308b\u3068\uff0c<\/p>\n
$$
\n\\frac{d}{dt}\\left\\{\\boldsymbol{g}\\cdot\\left(m\\boldsymbol{r}\\times\\frac{d\\boldsymbol{r}}{dt}\\right)\\right\\} = 0
\n$$<\/p>\n
\u3053\u306e\u5f0f\u306f\uff0c\u89d2\u904b\u52d5\u91cf $\\displaystyle \\boldsymbol{L} \\equiv m \\boldsymbol{r}\\times\\frac{d\\boldsymbol{r}}{dt}$ \u306e\u925b\u76f4\u6210\u5206\uff0c\u3064\u307e\u308a\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306b\u305d\u3063\u305f\u6210\u5206\u304c\u4fdd\u5b58\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n
\u904b\u52d5\u304c\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u7406\u7531<\/h4>\n
\u3053\u306e\u5f0f\u3092\u6210\u5206\u3067\u5177\u4f53\u7684\u306b\u66f8\u304d\u4e0b\u3059\u3068\uff08\u4e21\u8fba\u3092 $m g$ \u3067\u5272\u3063\u305f\u306e\u3061\uff09<\/p>\n
$$\\frac{d}{dt}\\left( x \\frac{dy}{dt} -y \\frac{dx}{dt}\\right) =
\n\\frac{d}{dt}\\left\\{
\nx^2 \\frac{d}{dt}\\left(\\frac{y}{x}\\right) \\right\\} = 0
\n$$<\/p>\n
\u6975\u5ea7\u6a19
\n$$\\boldsymbol{r} = (x, y, z) = (\\ell \\sin\\theta \\cos\\varphi, \\ell \\sin\\theta \\sin\\varphi, \\ell \\cos\\theta)
\n$$<\/p>\n
\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c<\/p>\n
$$\\frac{d}{dt}\\left\\{
\nx^2 \\frac{d}{dt}\\left(\\frac{y}{x}\\right) \\right\\} =
\n\\ell^2 \\frac{d}{dt} \\left(\\sin^2\\theta \\frac{d\\varphi}{dt} \\right) = 0
\n$$<\/p>\n
\u5fae\u5206\u3057\u3066\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4e2d\u8eab\u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3060\u304b\u3089\uff0c<\/p>\n
$$
\n\\sin^2\\theta \\frac{d\\varphi}{dt} = \\mbox{const.} \\equiv l_z
\n$$<\/p>\n
\u3053\u3053\u3067\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $t = 0$ \u3067 $\\displaystyle\\frac{d\\varphi}{dt} = 0$ \u3068\u3059\u308b\u3068\uff0c$l_z = 0$ \u3068\u306a\u308a\u5e38\u306b $\\displaystyle\\frac{d\\varphi}{dt} = 0$\uff0c\u3064\u307e\u308a $\\varphi = \\mbox{const.}$ \u3067\u3042\u308a\u3064\u3065\u3051\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u308b\u3002<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c\u5358\u3075\u308a\u3053\u306e\u904b\u52d5\u304c\uff08\u6163\u6027\u7cfb\u3067\u306f\uff09\u4e00\u5b9a\u306e\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u306e\u306f\uff0c\u4ee5\u4e0b\u306e\u7406\u7531\u306b\u3088\u308b\uff1a<\/p>\n
\n- \u89d2\u904b\u52d5\u91cf\u306e\u925b\u76f4\u6210\u5206\uff08\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306b\u5e73\u884c\u306a\u6210\u5206\uff09\u306f\u4fdd\u5b58\u3055\u308c\u308b\u3002\uff08\u4e0a\u8a18\u306e\u5834\u5408\u306f\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u5411\u304d\u306f $z$ \u65b9\u5411\u306a\u306e\u3067\uff0c\u89d2\u904b\u52d5\u91cf\u306e $z$ \u6210\u5206 $L_z = m \\ell^2 l_z$ \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3002\uff09<\/li>\n
- \u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $L_z = 0$ \u3068\u3059\u308b\u3068\uff0c\u305a\u30fc\u3063\u3068 $L_z = 0$ \u3067\u3042\u308a\u3064\u3065\u3051\u308b\u3002\uff08\u4e0a\u8a18\u306e\u5834\u5408\u306f\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $\\displaystyle \\frac{d\\varphi}{dt} = 0$ \u3068\u3059\u308b\u3068\uff0c\u305a\u30fc\u3063\u3068 $\\displaystyle \\frac{d\\varphi}{dt} = 0$\u3067\u3042\u308a\u3064\u3065\u3051\u308b\uff0c\u3064\u307e\u308a $\\varphi = \\mbox{const.}$\uff09<\/li>\n
- \u3057\u305f\u304c\u3063\u3066\uff0c\u904b\u52d5\u306f $\\varphi$ \u304c\u4e00\u5b9a\u306e\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3002<\/li>\n<\/ul>\n
\u3059\u306a\u308f\u3061\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066\uff0c$\\displaystyle y = 0, \\frac{dy}{dt} = 0$ \u3068\u3059\u308b\u3068\uff0c\u4ee5\u5f8c\u306e\u5358\u3075\u308a\u3053\u306e\u904b\u52d5\u306f $y = 0$ \u306e $xz$ \u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
\u6700\u7d42\u7684\u306a\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h4>\n
\u305d\u3053\u3067\u3053\u308c\u4ee5\u5f8c\u306f\uff0c$\\boldsymbol{r} = (x, y, z) = (\\ell\\sin\\theta, 0, \\ell\\cos\\theta)$ \u3068\u304a\u304f\u3002<\/p>\n
\u3059\u308b\u3068\uff0c$\\boldsymbol{r}$ \u3068\u5916\u7a4d\u3092\u3068\u3063\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u6b8b\u308a\u306e\u6210\u5206\u306e\u3046\u3061\uff0c$x$ \u6210\u5206\u306f $ 0 = 0$ \u3068\u306a\u308a\uff0c $y$ \u6210\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{d}{dt} \\left( z \\frac{dx}{dt} -x \\frac{dz}{dt} \\right) &=& -x g \\\\
\n\\frac{d}{dt} \\left\\{z^2 \\frac{d}{dt}\\left(\\frac{x}{z}\\right)\\right\\} &=& -x g \\\\
\n\\ell^2 \\frac{d}{dt} \\left\\{
\n\\cos^2\\theta \\frac{d}{dt}\\left( \\frac{\\sin\\theta}{\\cos\\theta}\\right)
\n\\right\\} &=& -\\ell g \\sin\\theta \\\\
\n\\ell^2 \\frac{d^2\\theta}{dt^2} =&=& -\\ell g \\sin\\theta
\n\\end{eqnarray}<\/p>\n
\u3067\u3042\u308b\u304b\u3089\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u6700\u7d42\u7684\u306b<\/p>\n
$$
\n\\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\sin\\theta
\n$$<\/p>\n
\u3068\u306a\u308b\u3002<\/p>\n
\u5358\u632f\u52d5\uff1a\u632f\u5e45\u304c\u5341\u5206\u306b\u5c0f\u3055\u3044\u5834\u5408<\/h3>\n
$|\\theta| \\ll 1$ \u306e\u5834\u5408\u306b\u306f\uff0c$\\sin\\theta \\simeq \\theta$ \u3068\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u306b\u3088\u308a\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$ \\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\theta$$<\/p>\n
\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u5358\u632f\u52d5\u3068\u306a\u308b\u3002<\/p>\n
\u5358\u632f\u52d5\u306e\u5468\u671f<\/h4>\n
\u5358\u632f\u52d5\u306e\u5468\u671f $\\tau_0$ \u306f<\/p>\n
$$ \\tau_0 = 2\\pi \\sqrt{\\frac{\\ell}{g}} $$<\/p>\n
\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u521d\u671f\u6761\u4ef6\u3067\u4e0e\u3048\u3089\u308c\u308b\u632f\u308c\u5e45\uff08\u632f\u5e45\uff09$\\theta_0$ \u306b\u4f9d\u5b58\u3057\u306a\u3044\u3002<\/p>\n
\u632f\u5e45\u304c\u5927\u304d\u3044\u5834\u5408\u306e\u5468\u671f\u306f\uff1f<\/h3>\n
\u632f\u5e45\u304c\u5927\u304d\u3044\uff0c\u3064\u307e\u308a $\\sin\\theta \\simeq \\theta$ \u3068\u8fd1\u4f3c\u3067\u304d\u306a\u3044\u5834\u5408\u306f\uff0c\u5358\u632f\u52d5\u3067\u306f\u306a\u3044\u904b\u52d5\u3092\u3059\u308b\u306f\u305a\u3060\u304b\u3089\uff0c\u3075\u308a\u3053\u306e\u5468\u671f\u306f\u4e00\u822c\u306b\u632f\u5e45\u306b\u4f9d\u5b58\u3059\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u3068\u8003\u3048\u3089\u308c\u308b\u3002<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f<\/p>\n
$$
\n\\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\sin\\theta
\n$$<\/p>\n
\u306f\u89e3\u6790\u7684\u306b\u306f\u89e3\u3051\u306a\u3044\u3002<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u7121\u6b21\u5143\u5316<\/h4>\n
\u89e3\u6790\u7684\u306b\u89e3\u3051\u306a\u3044\u5834\u5408\u306f\uff0c\u6570\u5024\u7684\u306b\u89e3\u304f\u3053\u3068\u306b\u306a\u308b\u3002\u305d\u3093\u306a\u3068\u304d\uff0c\u89e3\u304f\u3079\u304d\u65b9\u7a0b\u5f0f\u3092\u7121\u6b21\u5143\u5316\u3057\u3066\u304a\u304f\u3068\u3088\u3044\uff0c\u3068\u3044\u3046\u304b\uff0c\u7121\u6b21\u5143\u5316\u3059\u308b\u3079\u304d\u3067\u3042\u308b\u3002<\/p>\n
\u3053\u3053\u3067\u306f\uff0c\u5358\u632f\u52d5\u306e\u5834\u5408\u306e\u5468\u671f $\\tau_0$ \u306e2\u4e57\u3092\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u306b\u304b\u3051\u308b\u3002\u3059\u308b\u3068\uff0c\u53f3\u8fba\u306f<\/p>\n
$$-\\tau_0^2 \\times \\frac{g}{\\ell} \\sin\\theta = -4\\pi^2 \\sin\\theta$$<\/p>\n
\u3068\u306a\u308b\u3002\u307e\u305f\uff0c\u898f\u683c\u5316\u3055\u308c\u305f\u6642\u9593 $T$ \u3092<\/p>\n
$$ T \\equiv \\frac{t}{\\tau_0}$$<\/p>\n
\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3068<\/p>\n
$$ \\tau_0^2 \\frac{d^2}{dt^2} = \\frac{d^2}{dT^2}$$<\/p>\n
\u3068\u306a\u308b\u306e\u3067\uff0c\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
$$ \\frac{d^2\\theta}{dT^2} = -4\\pi^2 \\sin\\theta$$<\/p>\n
Runge-Kutta \u6cd5\u3067\u6570\u5024\u7684\u306b\u89e3\u304f\u305f\u3081\u306e\u6e96\u5099<\/h4>\n
<\/p>\n
\u5358\u632f\u308a\u5b50\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa<\/h3>\n
\u5358\u3075\u308a\u3053\u306e\u3072\u3082\u306e\u9577\u3055\u3092 $\\ell$\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u306e\u5927\u304d\u3055\u3092 $g$\uff0c\u925b\u76f4\u4e0b\u5411\u304d\u304b\u3089\u306e\u632f\u308c\u89d2\u3092 $\\theta$ \u3068\u3059\u308b\u3068\uff0c\u5358\u3075\u308a\u3053\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$
\n\\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\sin\\theta
\n$$<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n
<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h4>\n
\u5358\u3075\u308a\u3053\u306e\u5148\u7aef\u306e\u304a\u3082\u308a\u306f\uff0c\u3072\u3082\u306e\u5f35\u529b $\\boldsymbol{T}$ \u3068\u91cd\u529b $ m\\boldsymbol{g}$ \u3092\u53d7\u3051\u308b\u3002\u925b\u76f4\u4e0b\u5411\u304d\u3092 $+z$ \u65b9\u5411\u3068\u3059\u308b\u3068 $\\boldsymbol{g} = (0, 0, g)$ \u306f\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb, $g$ \u306f\u91cd\u529b\u52a0\u901f\u5ea6\u306e\u5927\u304d\u3055\u3067\u3042\u308b\u3002<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$m \\frac{d^2\\boldsymbol{r}}{dt^2} = m\\boldsymbol{g} + \\boldsymbol{T}$$<\/p>\n
\u4fdd\u5b58\u91cf\uff08\u89d2\u904b\u52d5\u91cf\u306e\u925b\u76f4\u6210\u5206\uff09<\/h4>\n
\u3053\u306e\u5f0f\u306e\u4e21\u8fba\u306b\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306e\u5916\u7a4d\u3092\u304b\u3051\u308b\u3002$\\boldsymbol{T} \\propto -\\boldsymbol{r}$ \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068\uff0c$\\boldsymbol{r}\\times \\boldsymbol{T} = \\boldsymbol{0}$ \u3067\u3042\u308b\u304b\u3089\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
$$\\frac{d}{dt}\\left(m \\boldsymbol{r}\\times\\frac{d\\boldsymbol{r}}{dt}\\right) = m \\boldsymbol{r}\\times\\boldsymbol{g}$$<\/p>\n
\u3053\u306e\u4e21\u8fba\u306b\u4e00\u5b9a\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb $\\boldsymbol{g}$ \u306e\u5185\u7a4d\u3092\u304b\u3051\u3066\u3084\u308b\u3068\uff0c<\/p>\n
$$
\n\\frac{d}{dt}\\left\\{\\boldsymbol{g}\\cdot\\left(m\\boldsymbol{r}\\times\\frac{d\\boldsymbol{r}}{dt}\\right)\\right\\} = 0
\n$$<\/p>\n
\u3053\u306e\u5f0f\u306f\uff0c\u89d2\u904b\u52d5\u91cf $\\displaystyle \\boldsymbol{L} \\equiv m \\boldsymbol{r}\\times\\frac{d\\boldsymbol{r}}{dt}$ \u306e\u925b\u76f4\u6210\u5206\uff0c\u3064\u307e\u308a\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306b\u305d\u3063\u305f\u6210\u5206\u304c\u4fdd\u5b58\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/p>\n
\u904b\u52d5\u304c\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u7406\u7531<\/h4>\n
\u3053\u306e\u5f0f\u3092\u6210\u5206\u3067\u5177\u4f53\u7684\u306b\u66f8\u304d\u4e0b\u3059\u3068\uff08\u4e21\u8fba\u3092 $m g$ \u3067\u5272\u3063\u305f\u306e\u3061\uff09<\/p>\n
$$\\frac{d}{dt}\\left( x \\frac{dy}{dt} -y \\frac{dx}{dt}\\right) =
\n\\frac{d}{dt}\\left\\{
\nx^2 \\frac{d}{dt}\\left(\\frac{y}{x}\\right) \\right\\} = 0
\n$$<\/p>\n
\u6975\u5ea7\u6a19
\n$$\\boldsymbol{r} = (x, y, z) = (\\ell \\sin\\theta \\cos\\varphi, \\ell \\sin\\theta \\sin\\varphi, \\ell \\cos\\theta)
\n$$<\/p>\n
\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c<\/p>\n
$$\\frac{d}{dt}\\left\\{
\nx^2 \\frac{d}{dt}\\left(\\frac{y}{x}\\right) \\right\\} =
\n\\ell^2 \\frac{d}{dt} \\left(\\sin^2\\theta \\frac{d\\varphi}{dt} \\right) = 0
\n$$<\/p>\n
\u5fae\u5206\u3057\u3066\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4e2d\u8eab\u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3060\u304b\u3089\uff0c<\/p>\n
$$
\n\\sin^2\\theta \\frac{d\\varphi}{dt} = \\mbox{const.} \\equiv l_z
\n$$<\/p>\n
\u3053\u3053\u3067\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $t = 0$ \u3067 $\\displaystyle\\frac{d\\varphi}{dt} = 0$ \u3068\u3059\u308b\u3068\uff0c$l_z = 0$ \u3068\u306a\u308a\u5e38\u306b $\\displaystyle\\frac{d\\varphi}{dt} = 0$\uff0c\u3064\u307e\u308a $\\varphi = \\mbox{const.}$ \u3067\u3042\u308a\u3064\u3065\u3051\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u308b\u3002<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c\u5358\u3075\u308a\u3053\u306e\u904b\u52d5\u304c\uff08\u6163\u6027\u7cfb\u3067\u306f\uff09\u4e00\u5b9a\u306e\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u306e\u306f\uff0c\u4ee5\u4e0b\u306e\u7406\u7531\u306b\u3088\u308b\uff1a<\/p>\n
- \n
- \u89d2\u904b\u52d5\u91cf\u306e\u925b\u76f4\u6210\u5206\uff08\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306b\u5e73\u884c\u306a\u6210\u5206\uff09\u306f\u4fdd\u5b58\u3055\u308c\u308b\u3002\uff08\u4e0a\u8a18\u306e\u5834\u5408\u306f\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u5411\u304d\u306f $z$ \u65b9\u5411\u306a\u306e\u3067\uff0c\u89d2\u904b\u52d5\u91cf\u306e $z$ \u6210\u5206 $L_z = m \\ell^2 l_z$ \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3002\uff09<\/li>\n
- \u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $L_z = 0$ \u3068\u3059\u308b\u3068\uff0c\u305a\u30fc\u3063\u3068 $L_z = 0$ \u3067\u3042\u308a\u3064\u3065\u3051\u308b\u3002\uff08\u4e0a\u8a18\u306e\u5834\u5408\u306f\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $\\displaystyle \\frac{d\\varphi}{dt} = 0$ \u3068\u3059\u308b\u3068\uff0c\u305a\u30fc\u3063\u3068 $\\displaystyle \\frac{d\\varphi}{dt} = 0$\u3067\u3042\u308a\u3064\u3065\u3051\u308b\uff0c\u3064\u307e\u308a $\\varphi = \\mbox{const.}$\uff09<\/li>\n
- \u3057\u305f\u304c\u3063\u3066\uff0c\u904b\u52d5\u306f $\\varphi$ \u304c\u4e00\u5b9a\u306e\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3002<\/li>\n<\/ul>\n
\u3059\u306a\u308f\u3061\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066\uff0c$\\displaystyle y = 0, \\frac{dy}{dt} = 0$ \u3068\u3059\u308b\u3068\uff0c\u4ee5\u5f8c\u306e\u5358\u3075\u308a\u3053\u306e\u904b\u52d5\u306f $y = 0$ \u306e $xz$ \u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
\u6700\u7d42\u7684\u306a\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h4>\n
\u305d\u3053\u3067\u3053\u308c\u4ee5\u5f8c\u306f\uff0c$\\boldsymbol{r} = (x, y, z) = (\\ell\\sin\\theta, 0, \\ell\\cos\\theta)$ \u3068\u304a\u304f\u3002<\/p>\n
\u3059\u308b\u3068\uff0c$\\boldsymbol{r}$ \u3068\u5916\u7a4d\u3092\u3068\u3063\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u6b8b\u308a\u306e\u6210\u5206\u306e\u3046\u3061\uff0c$x$ \u6210\u5206\u306f $ 0 = 0$ \u3068\u306a\u308a\uff0c $y$ \u6210\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{d}{dt} \\left( z \\frac{dx}{dt} -x \\frac{dz}{dt} \\right) &=& -x g \\\\
\n\\frac{d}{dt} \\left\\{z^2 \\frac{d}{dt}\\left(\\frac{x}{z}\\right)\\right\\} &=& -x g \\\\
\n\\ell^2 \\frac{d}{dt} \\left\\{
\n\\cos^2\\theta \\frac{d}{dt}\\left( \\frac{\\sin\\theta}{\\cos\\theta}\\right)
\n\\right\\} &=& -\\ell g \\sin\\theta \\\\
\n\\ell^2 \\frac{d^2\\theta}{dt^2} =&=& -\\ell g \\sin\\theta
\n\\end{eqnarray}<\/p>\n\u3067\u3042\u308b\u304b\u3089\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u6700\u7d42\u7684\u306b<\/p>\n
$$
\n\\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\sin\\theta
\n$$<\/p>\n\u3068\u306a\u308b\u3002<\/p>\n
\u5358\u632f\u52d5\uff1a\u632f\u5e45\u304c\u5341\u5206\u306b\u5c0f\u3055\u3044\u5834\u5408<\/h3>\n
$|\\theta| \\ll 1$ \u306e\u5834\u5408\u306b\u306f\uff0c$\\sin\\theta \\simeq \\theta$ \u3068\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u306b\u3088\u308a\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$ \\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\theta$$<\/p>\n
\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u5358\u632f\u52d5\u3068\u306a\u308b\u3002<\/p>\n
\u5358\u632f\u52d5\u306e\u5468\u671f<\/h4>\n
\u5358\u632f\u52d5\u306e\u5468\u671f $\\tau_0$ \u306f<\/p>\n
$$ \\tau_0 = 2\\pi \\sqrt{\\frac{\\ell}{g}} $$<\/p>\n
\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u521d\u671f\u6761\u4ef6\u3067\u4e0e\u3048\u3089\u308c\u308b\u632f\u308c\u5e45\uff08\u632f\u5e45\uff09$\\theta_0$ \u306b\u4f9d\u5b58\u3057\u306a\u3044\u3002<\/p>\n
\u632f\u5e45\u304c\u5927\u304d\u3044\u5834\u5408\u306e\u5468\u671f\u306f\uff1f<\/h3>\n
\u632f\u5e45\u304c\u5927\u304d\u3044\uff0c\u3064\u307e\u308a $\\sin\\theta \\simeq \\theta$ \u3068\u8fd1\u4f3c\u3067\u304d\u306a\u3044\u5834\u5408\u306f\uff0c\u5358\u632f\u52d5\u3067\u306f\u306a\u3044\u904b\u52d5\u3092\u3059\u308b\u306f\u305a\u3060\u304b\u3089\uff0c\u3075\u308a\u3053\u306e\u5468\u671f\u306f\u4e00\u822c\u306b\u632f\u5e45\u306b\u4f9d\u5b58\u3059\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u3068\u8003\u3048\u3089\u308c\u308b\u3002<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f<\/p>\n
$$
\n\\frac{d^2\\theta}{dt^2} = -\\frac{g}{\\ell} \\sin\\theta
\n$$<\/p>\n\u306f\u89e3\u6790\u7684\u306b\u306f\u89e3\u3051\u306a\u3044\u3002<\/p>\n
\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u7121\u6b21\u5143\u5316<\/h4>\n
\u89e3\u6790\u7684\u306b\u89e3\u3051\u306a\u3044\u5834\u5408\u306f\uff0c\u6570\u5024\u7684\u306b\u89e3\u304f\u3053\u3068\u306b\u306a\u308b\u3002\u305d\u3093\u306a\u3068\u304d\uff0c\u89e3\u304f\u3079\u304d\u65b9\u7a0b\u5f0f\u3092\u7121\u6b21\u5143\u5316\u3057\u3066\u304a\u304f\u3068\u3088\u3044\uff0c\u3068\u3044\u3046\u304b\uff0c\u7121\u6b21\u5143\u5316\u3059\u308b\u3079\u304d\u3067\u3042\u308b\u3002<\/p>\n
\u3053\u3053\u3067\u306f\uff0c\u5358\u632f\u52d5\u306e\u5834\u5408\u306e\u5468\u671f $\\tau_0$ \u306e2\u4e57\u3092\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u306b\u304b\u3051\u308b\u3002\u3059\u308b\u3068\uff0c\u53f3\u8fba\u306f<\/p>\n
$$-\\tau_0^2 \\times \\frac{g}{\\ell} \\sin\\theta = -4\\pi^2 \\sin\\theta$$<\/p>\n
\u3068\u306a\u308b\u3002\u307e\u305f\uff0c\u898f\u683c\u5316\u3055\u308c\u305f\u6642\u9593 $T$ \u3092<\/p>\n
$$ T \\equiv \\frac{t}{\\tau_0}$$<\/p>\n
\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3068<\/p>\n
$$ \\tau_0^2 \\frac{d^2}{dt^2} = \\frac{d^2}{dT^2}$$<\/p>\n
\u3068\u306a\u308b\u306e\u3067\uff0c\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n
$$ \\frac{d^2\\theta}{dT^2} = -4\\pi^2 \\sin\\theta$$<\/p>\n
Runge-Kutta \u6cd5\u3067\u6570\u5024\u7684\u306b\u89e3\u304f\u305f\u3081\u306e\u6e96\u5099<\/h4>\n