\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5149\u306e\u7d4c\u8def\u3092\u6c7a\u3081\u308b\u5f0f\u30922\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\u89e3\u304f\u3002<\/p>\n
<\/p>\n
\\begin{eqnarray}
\n\\left(\\frac{du}{d\\phi}\\right)^2
\n&=& \\frac{1}{b^2} -u^2 + r_g \\left(\\,u^3\u00a0 -\\frac{1}{b^3}\\right)
\n\\end{eqnarray}<\/p>\n
\u4e21\u8fba\u3092 $\\phi$ \u3067\u5fae\u5206\u3057\u30662\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3059\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n2\\frac{du}{d\\phi}\\,\\frac{d^2u}{d\\phi^2}
\n&=&\u00a0 -2 u\\,\u00a0 \\frac{du}{d\\phi}+ 3 r_g \\,u^2 \\, \\frac{du}{d\\phi} \\\\
\n\\therefore\\ \\ \\frac{d^2u}{d\\phi^2} &=& -u + \\frac{3}{2} r_g \\, u^2
\n\\end{eqnarray}<\/p>\n
1\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u3068\u304d\u306b\u306f1\u500b\u306e\u7a4d\u5206\u5b9a\u6570\u3092\u6c7a\u3081\u308b\u305f\u3081\u306e\u521d\u671f\u6761\u4ef6\u306f\u3072\u3068\u3064\u3067\u3088\u304f\u3066\uff0c$\\displaystyle \\phi = \\frac{\\pi}{2}$ \u306e\u3068\u304d\uff0c$\\displaystyle u = \\frac{1}{b}$ \u3068\u3057\u305f\u30022\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5834\u5408\u306f\uff0c2\u500b\u306e\u7a4d\u5206\u5b9a\u6570\u3092\u6c7a\u3081\u308b\u305f\u3081\u306b\u521d\u671f\u6761\u4ef6\u3082\u3075\u305f\u3064\u5fc5\u8981\u3068\u306a\u308b\u304b\u3089\uff0c\u3082\u3068\u306e1\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u304b\u3089\u4ee5\u4e0b\u3092\u63a1\u7528\u3059\u308b\u3002<\/p>\n
$r_g$ \u304c\u304b\u304b\u3063\u305f\u9805\u3092\u7121\u8996\u3057\u305f\u3068\u304d\u306e\u89e3\u3092 $u_0$ \u3068\u304a\u304f\u3068<\/p>\n
\\begin{eqnarray}
\n\\frac{d^2 u_0}{d\\phi^2} &=& -u_0
\n\\end{eqnarray}<\/p>\n
\u521d\u671f\u6761\u4ef6 $\\displaystyle \\phi = \\frac{\\pi}{2}$ \u306e\u3068\u304d\uff0c$\\displaystyle u_0 = \\frac{1}{b}$\uff0c$\\displaystyle \\frac{du_0}{d\\phi} = 0$ \u3088\u308a<\/p>\n
$$u_0 = \\frac{\\sin \\phi}{b}$$<\/p>\n
$\\displaystyle\u00a0 u = u_0 + \\frac{r_g}{b} u_1$ \u3068\u304a\u3044\u3066\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3057\uff0c$r_g$ \u306e1\u6b21\u306e\u9805\u3092\u53d6\u308a\u51fa\u3059\u3068<\/p>\n
$$\\frac{d^2 u_1}{d\\phi^2} + u_1 = \\frac{3}{2 b} \\sin^2 \\phi$$<\/p>\n
\u3053\u306e\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\uff08\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u3068\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u306e\u548c\uff09\u306b\u521d\u671f\u6761\u4ef6\u3092\u8ab2\u3059\u3002<\/p>\n
$u$ \u5168\u4f53\u306b\u5bfe\u3059\u308b\u521d\u671f\u6761\u4ef6\u306f $\\displaystyle \\phi = \\frac{\\pi}{2}$ \u306e\u3068\u304d\uff0c$\\displaystyle u = \\frac{1}{b}$\uff0c$\\displaystyle \\frac{du}{d\\phi} = 0$ \u3067\u3042\u308a\uff0c\u3053\u308c\u306f $u_1$ \u306b\u5bfe\u3057\u3066\u306f\uff0c$\\displaystyle u_1 = 0, \\frac{d u_1}{d\\phi} = 0$ \u3068\u306a\u308b\u304b\u3089\uff0c\u89e3\u306f<\/p>\n
$$ u_1 = \\frac{1}{2 b} \\left( 2 -\\sin \\phi -\\sin^2 \\phi \\right)$$<\/p>\n
\u3057\u305f\u304c\u3063\u3066 $r_g$ \u306e1\u6b21\u307e\u3067\u306e\u89e3\u306f<\/p>\n
$$u = u_0 + \\frac{r_g}{b} u_1 = \\frac{\\sin\\phi}{b} + \\frac{r_g}{2 b^2} \\left( 2 -\\sin\\phi -\\sin^2\\phi\\right)$$<\/p>\n
eq<\/span>:<\/span> '<\/span>diff<\/span>(<\/span>u1<\/span>, phi<\/span>, 2<\/span>)<\/span> +<\/span> u1<\/span> =<\/span> 3\/<\/span>(<\/span>2*<\/span>b<\/span>)<\/span> *<\/span> sin<\/span>(<\/span>phi<\/span>)<\/span>**<\/span>2<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\nOut[1]:<\/div>\n\\[\\tag{${\\it \\%o}_{1}$}\\frac{d^2}{d\\,\\varphi^2}\\,u_{1}+u_{1}=\\frac{3\\,\\sin ^2\\varphi}{2\\,b}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\nIn\u00a0[2]:<\/div>\n\n\n\nsol<\/span>:<\/span> ode2<\/span>(<\/span>eq<\/span>, u1<\/span>, phi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\nOut[2]:<\/div>\n\\[\\tag{${\\it \\%o}_{2}$}u_{1}=\\frac{\\cos \\left(2\\,\\varphi\\right)+3}{4\\,b}+{\\it \\%k}_{1}\\,\\sin \\varphi+{\\it \\%k}_{2}\\,\\cos \\varphi\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n\u521d\u671f\u6761\u4ef6 $\\displaystyle \\phi = \\frac{\\pi}{2}$ \u3067 $\\displaystyle u_1 = 0, \\frac{d u_1}{d\\phi} = 0$ \u3092\u8ab2\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[3]:<\/div>\n\n\n\nsol1<\/span>:<\/span> ic2<\/span>(<\/span>sol<\/span>, phi<\/span> =<\/span> %pi<\/span>\/<\/span>2<\/span>, u1<\/span> =<\/span> 0<\/span>, '<\/span>diff<\/span>(<\/span>u1<\/span>, phi<\/span>)<\/span> =<\/span> 0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\nOut[3]:<\/div>\n\\[\\tag{${\\it \\%o}_{3}$}u_{1}=\\frac{\\cos \\left(2\\,\\varphi\\right)+3}{4\\,b}-\\frac{\\sin \\varphi}{2\\,b}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n$\\cos 2 \\phi \\Rightarrow 1 – 2 \\sin^2 \\phi$ \u3068\u7f6e\u304d\u63db\u3048\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[4]:<\/div>\n\n\n\nev<\/span>(<\/span>sol1<\/span>, cos<\/span>(<\/span>2*<\/span>phi<\/span>)<\/span>=<\/span>1<\/span> -<\/span> 2*<\/span>sin<\/span>(<\/span>phi<\/span>)<\/span>**<\/span>2<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\nOut[4]:<\/div>\n\\[\\tag{${\\it \\%o}_{4}$}u_{1}=-\\frac{\\sin ^2\\varphi+\\sin \\varphi-2}{2\\,b}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n\n<\/div>\n\n\nSymPy \u3067\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u307f\u308b<\/h3>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h4>\n<\/div>\n<\/div>\n<\/div>\n\n\nIn\u00a0[1]:<\/div>\n\n\n\nfrom<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\nIn\u00a0[2]:<\/div>\n\n\n\nu1<\/span> =<\/span> Function<\/span>(<\/span>'u1'<\/span>)<\/span>\r\n\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>Derivative<\/span>(<\/span>u1<\/span>(<\/span>phi<\/span>),<\/span> phi<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> u1<\/span>(<\/span>phi<\/span>),<\/span> \r\n Rational<\/span>(<\/span>3<\/span>,<\/span>2<\/span>)<\/span> *<\/span> sin<\/span>(<\/span>phi<\/span>)<\/span>**<\/span>2<\/span>\/<\/span>b<\/span>\r\n)<\/span>\r\neq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\nOut[2]:<\/div>\n$\\displaystyle u_{1}{\\left(\\phi \\right)} + \\frac{d^{2}}{d \\phi^{2}} u_{1}{\\left(\\phi \\right)} = \\frac{3 \\sin^{2}{\\left(\\phi \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\nIn\u00a0[3]:<\/div>\n\n\n\ndsolve<\/span>(<\/span>eq<\/span>,<\/span> u1<\/span>(<\/span>phi<\/span>),<\/span> \r\n ics<\/span> =<\/span> {<\/span>u1<\/span>(<\/span>phi<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>phi<\/span>,<\/span> pi<\/span>\/<\/span>2<\/span>):<\/span>0<\/span>,<\/span> # u1(pi\/2) = 0<\/span>\r\n diff<\/span>(<\/span>u1<\/span>(<\/span>