\u5358\u632f\u308a\u5b50\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u5247\u304b\u3089\uff0c\u6570\u5024\u7a4d\u5206\u306b\u3088\u3063\u3066\u5468\u671f\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u6e96\u5099\u3002<\/p>\n
<\/p>\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
<\/p>\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f<\/p>\n
$$ \\frac{d^2\\theta}{dT^2} = -4\\pi^2 \\sin\\theta$$<\/p>\n
\u306e\u4e21\u8fba\u306b $\\displaystyle \\frac{d\\theta}{dT}$ \u3092\u304b\u3051\u3066\u6574\u7406\u3059\u308b\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\frac{d\\theta}{dT}\\frac{d^2\\theta}{dT^2}+4\\pi^2 \\sin\\theta\u00a0 \\frac{d\\theta}{dT}&=& 0 \\\\
\n\\frac{d}{dT}\\left\\{
\n\\frac{1}{2}\\left( \\frac{d\\theta}{dT} \\right)^2 -4\\pi^2 \\cos\\theta
\n\\right\\} &=& 0 \\\\
\n\\therefore\\ \\ \\frac{1}{2}\\left( \\frac{d\\theta}{dT} \\right)^2 -4\\pi^2 \\cos\\theta
\n&=& \\mbox{const.} \\equiv \\varepsilon
\n\\end{eqnarray}<\/p>\n
\u3064\u307e\u308a\uff0c<\/p>\n
$$
\n\\varepsilon = \\frac{1}{2}\\left( \\frac{d\\theta}{dT} \\right)^2 -4\\pi^2 \\cos\\theta
\n$$<\/p>\n
\u3068\u3044\u3046\u4fdd\u5b58\u91cf\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3053\u308c\u306f\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u5247\u306b\u76f8\u5f53\u3059\u308b\u3002<\/p>\n
$T = 0$ \u3067 $\\displaystyle \\theta = \\theta_0, \\frac{d\\theta}{dT} = 0$ \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u3092\u8ab2\u3059\u3068\uff0c\u5b9a\u6570 $\\varepsilon$ \u306e\u5024\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u307e\u308b\u3002
\n$$ \\varepsilon = -4\\pi^2 \\cos\\theta_0$$<\/p>\n
\u3064\u307e\u308a\uff0c$\\theta = \\theta_0$ \u304b\u3089 $\\theta = 0$ \u307e\u3067 $\\frac{d\\theta}{dT} < 0$ \u306e\u72b6\u6cc1\u3092\u8003\u3048\u308b\u3053\u3068\u306b\u3057\u3066<\/p>\n
\\begin{eqnarray}
\n\\frac{1}{2}\\left( \\frac{d\\theta}{dT} \\right)^2 &=& 4\\pi^2 \\left( \\cos\\theta-\\cos\\theta_0\\right)\\\\
\n\\frac{d\\theta}{dT} &=& -2\\pi \\sqrt{2(\\cos\\theta-\\cos\\theta_0)} \\\\
\n-\\frac{1}{2\\pi} \\frac{d\\theta}{\\sqrt{2(\\cos\\theta-\\cos\\theta_0)}} &=& dT
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u306f\u5909\u6570\u5206\u96e2\u5f62\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3042\u308a\uff0c$\\theta$ \u304c $\\theta_0$ \u304b\u3089 $0$ \u307e\u3067\u306e\u6642\u9593\u306e4\u500d\u304c\u5468\u671f $T_p$ \u306b\u306a\u308b\u3053\u3068\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\n-\\frac{1}{2\\pi} \\int^0_{\\theta_0} \\frac{d\\theta}{\\sqrt{2(\\cos\\theta-\\cos\\theta_0)}} &=& \\int_0^{\\frac{T_p}{4}} dT
\n\\end{eqnarray}<\/p>\n
\u3059\u306a\u308f\u3061<\/p>\n
$$
\nT_p = \\frac{2}{\\pi} \\int_0^{\\theta_0} \\frac{1}{\\sqrt{2(\\cos\\theta-\\cos\\theta_0)}}d\\theta
\n$$<\/p>\n
\u3053\u306e\u7a4d\u5206\u306f\u89e3\u6790\u7684\u306b\u306f\u89e3\u3051\u306a\u3044\u306e\u3067\u6570\u5024\u7a4d\u5206\u3059\u308c\u3070\u3044\u3044\u3068\u601d\u3044\u307e\u3059\u304c\uff0c\u3053\u306e\u307e\u307e\u306e\u5f62\u3060\u3068\uff0c\u7a4d\u5206\u533a\u9593\u306e\u4e0a\u9650\u3067\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u3063\u3066\u3057\u307e\u3044\uff0c\u3064\u307e\u308a\u306f\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u767a\u6563\u3057\u3066\u3057\u307e\u3046\u306e\u3067\uff0c\u5927\u5909\u306a\u3053\u3068\u304c\u304a\u3053\u308a\u305d\u3046…<\/p>\n
quad_qags()<\/code> \u3084 Python \u306e scipy.integrate.quad()<\/code> \u306a\u3089\u7121\u554f\u984c<\/h3>\n… \u306a\u3069\u3068\uff0c\u3044\u308d\u3044\u308d\u5fc3\u914d\u3057\u307e\u3057\u305f\u304c\uff0c\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u767a\u6563\u3059\u308b\u304b\u3089\u3068\u3044\u3063\u3066\u7a4d\u5206\u305d\u306e\u3082\u306e\u304c\u767a\u6563\u3059\u308b\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u6570\u5024\u7a4d\u5206\u3092\u3059\u308b\u95a2\u6570\u3067\u3042\u308b Maxima \u306e quad_qags()<\/code> \uff08romberg()<\/code> \u306f\u30c0\u30e1\uff09\u3084\uff0cPython \u306e scipy.integrate.quad()<\/code> \u306f\uff0c$\\theta = \\theta_0$ \u3067\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u767a\u6563\u3057\u3066\u3057\u307e\u3046\u3088\u3046\u306a\u5834\u5408\u3067\u3082\uff0c\u4f55\u4e8b\u3082\u306a\u304b\u3063\u305f\u3088\u3046\u306b\u6570\u5024\u7a4d\u5206\u3057\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n\u7b2c1\u7a2e\u5b8c\u5168\u6955\u5186\u7a4d\u5206<\/h3>\n
\u3053\u3053\u3067\u306f\uff0c\u4e0a\u8a18\u306e\u7a4d\u5206\u304c\u5909\u6570\u5909\u63db\u306b\u3088\u3063\u3066\uff0c\u7b2c1\u7a2e\u5b8c\u5168\u6955\u5186\u7a4d\u5206<\/strong><\/span> $K(k)$ \u3092\u4f7f\u3063\u3066
\n\\begin{eqnarray}
\n\\int_0^{\\theta_0} \\frac{1}{\\sqrt{2(\\cos\\theta-\\cos\\theta_0)}} d\\theta
\n&=& K(k) \\\\&\\equiv&
\n\\int_0^{\\pi\/2} \\frac{dt}{\\sqrt{1 -k^2 \\sin^2 t}}, \\quad k = \\sin\\frac{\\theta_0}{2}
\n\\end{eqnarray}<\/p>\n\u3067\u66f8\u3051\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u3053\u306e\u5f62\u3060\u3068\uff0c$k^2 < 1$ \u306a\u3089\u7a4d\u5206\u533a\u9593\u5185\u3067\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u767a\u6563\u3059\u308b\u3053\u3068\u3082\u306a\u304f\uff0c\u6570\u5024\u7a4d\u5206\u306b\u56f0\u308b\u3053\u3068\u3082\u306a\u3044\u3002<\/p>\n
\u307e\u305a\uff0c\u5206\u6bcd\u306f\u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066<\/p>\n
\\begin{eqnarray}
\n\\sqrt{2 (\\cos\\theta-\\cos\\theta_0)} &=&
\n\\sqrt{2 \\left\\{ \\left(1-2\\sin^2\\frac{\\theta}{2}\\right)
\n-\\left(1-2\\sin^2\\frac{\\theta_0}{2}\\right)
\n\\right\\} }\\\\
\n&=& 2\\sqrt{\\sin^2\\frac{\\theta_0}{2} -\\sin^2\\frac{\\theta}{2}}
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c<\/p>\n
$$\\displaystyle \\sin\\frac{\\theta_0}{2} \\equiv k, \\quad
\n\\sin\\frac{\\theta}{2} \\equiv k \\sin t
\n$$<\/p>\n
\u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n2\\sqrt{\\sin^2\\frac{\\theta_0}{2} -\\sin^2\\frac{\\theta}{2}}
\n&=& 2 k \\sqrt{1 -\\sin^2 t} \\\\
\n&=& 2 k \\cos t
\n\\end{eqnarray}<\/p>\n
\u4e00\u65b9\uff0c\u5206\u5b50\u306f<\/p>\n
$$ \\sin\\frac{\\theta}{2} \\equiv k \\sin t $$<\/p>\n
\u306e\u5fae\u5206\u3092\u3068\u308b\u3053\u3068\u306b\u3088\u308a\uff0c<\/p>\n
$$ \\cos\\frac{\\theta}{2} \\frac{d\\theta}{2} = k \\cos t \\, dt$$<\/p>\n
$$
\n\\therefore \\ d\\theta = \\frac{2 k \\cos t}{\\cos\\frac{\\theta}{2}} dt
\n= \\frac{2 k \\cos t}{\\sqrt{1-\\sin^2\\frac{\\theta}{2}}} dt
\n= \\frac{2 k \\cos t}{\\sqrt{1-k^2 \\sin^2 t}} dt
\n$$<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n
\\begin{eqnarray}
\n\\int_0^{\\theta_0} \\frac{1}{\\sqrt{2(\\cos\\theta-\\cos\\theta_0)}} d\\theta
\n&=&\\int_0^{\\pi\/2}
\n\\frac{1}{2 k \\cos t} \\frac{2 k \\cos t}{\\sqrt{1-k^2 \\sin^2 t}} dt\\\\
\n&=&\\int_0^{\\pi\/2} \\frac{dt}{\\sqrt{1-k^2 \\sin^2 t}}\\\\
\n&\\equiv& K(k)
\n\\end{eqnarray}<\/p>\n
\u7b2c1\u7a2e\u5b8c\u5168\u6955\u5186\u7a4d\u5206<\/strong><\/span> $K(k)$ \u3092\u4f7f\u3046\u3068\uff0c\u632f\u5e45 $\\theta_0$ \u306e\u5358\u632f\u308a\u5b50\u306e\u5468\u671f $T_p$ \u306f<\/p>\n\\begin{eqnarray}
\nT_p(\\theta_0) &=&
\n\\frac{2}{\\pi} \\int_{0}^{\\theta_0} \\frac{1}{\\sqrt{2(\\cos\\theta-\\cos\\theta_0)}}d\\theta \\\\
\n&=& \\frac{2}{\\pi} K(k) \\\\
\n&=& \\frac{2}{\\pi}\\int_0^{\\pi\/2} \\frac{dt}{\\sqrt{1-k^2 \\sin^2 t}}, \\quad k \\equiv \\sin\\frac{\\theta_0}{2}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5358\u632f\u308a\u5b50\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58\u5247\u304b\u3089\uff0c\u6570\u5024\u7a4d\u5206\u306b\u3088\u3063\u3066\u5468\u671f\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u6e96\u5099\u3002<\/p>