{"id":7803,"date":"2024-02-28T12:30:53","date_gmt":"2024-02-28T03:30:53","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7803"},"modified":"2024-06-11T15:55:14","modified_gmt":"2024-06-11T06:55:14","slug":"%e5%8f%82%e8%80%83%ef%bc%9asympy-%e3%81%a7%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%83%bb%e5%91%a8%ef%bc%8c%e5%9b%9e%e8%bb%a2%e6%a5%95%e5%86%86%e4%bd%93%e3%81%ae%e8%a1%a8%e9%9d%a2%e7%a9%8d","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8\/%e5%8f%82%e8%80%83%ef%bc%9asympy-%e3%81%a7%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%83%bb%e5%91%a8%ef%bc%8c%e5%9b%9e%e8%bb%a2%e6%a5%95%e5%86%86%e4%bd%93%e3%81%ae%e8%a1%a8%e9%9d%a2%e7%a9%8d\/","title":{"rendered":"\u53c2\u8003\uff1aSymPy \u3067\u6955\u5186\u306e\u9762\u7a4d\u30fb\u5468\uff0c\u56de\u8ee2\u6955\u5186\u4f53\u306e\u8868\u9762\u7a4d\u30fb\u4f53\u7a4d"},"content":{"rendered":"

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\u5fc5\u8981\u306a\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n# SymPy Plotting Backends (SPB) <\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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mathtext.fontset<\/code> \u306e\u8a2d\u5b9a<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u305f\u3081\u3067\u306f\u306a\u304f\u30c7\u30d5\u30a9\u30eb\u30c8\u8a2d\u5b9a\u306e\u305f\u3081<\/span>\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\n\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'mathtext.fontset'<\/span>]<\/span> =<\/span> 'cm'<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u9762\u7a4d<\/h3>\n
SymPy Plotting Backends \u3067\u9577\u534a\u5f84 $a$\uff0c\u5358\u534a\u5f84 $b$ \u306e\u6955\u5186\u3092\u63cf\u304f\u4f8b\u3002\uff08\u4e00\u90e8\u304c\u5fae\u5999\u306b\u30ab\u30af\u30ab\u30af\u3059\u308b\u306e\u3067\uff0cn<\/code> \u3092\u5927\u304d\u3081\u306b\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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a<\/span> =<\/span> 2<\/span>\r\nb<\/span> =<\/span> 1<\/span>\r\np<\/span> =<\/span> plot_geometry<\/span>(<\/span>\r\n    Ellipse<\/span>(<\/span>Point<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>),<\/span> hradius<\/span>=<\/span>a<\/span>,<\/span> vradius<\/span>=<\/span>b<\/span>),<\/span>\r\n    {<\/span>\"facecolor\"<\/span>:<\/span> \"yellow\"<\/span>,<\/span> \"edgecolor\"<\/span>:<\/span>\"blue\"<\/span>,<\/span> \"lw\"<\/span>:<\/span>2<\/span>},<\/span> \r\n    aspect<\/span> =<\/span> 'equal'<\/span>,<\/span> size<\/span>=<\/span>(<\/span>6.4<\/span>,<\/span>6.4<\/span>),<\/span> n<\/span>=<\/span>500<\/span>,<\/span> show<\/span>=<\/span>False<\/span>);<\/span>\r\n\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> p<\/span>.<\/span>fig<\/span>,<\/span> p<\/span>.<\/span>ax<\/span>\r\nfig<\/span>.<\/span>tight_layout<\/span>()<\/span>\r\n\r\nax<\/span>.<\/span>grid<\/span>(<\/span>linestyle<\/span> =<\/span> 'dotted'<\/span>)<\/span>\r\n# \u8868\u793a\u7bc4\u56f2<\/span>\r\nax<\/span>.<\/span>set_xlim<\/span>(<\/span>-<\/span>a<\/span>-<\/span>0.5<\/span>,<\/span> a<\/span>+<\/span>0.5<\/span>)<\/span>\r\nax<\/span>.<\/span>set_ylim<\/span>(<\/span>-<\/span>b<\/span>-<\/span>0.5<\/span>,<\/span> b<\/span>+<\/span>0.5<\/span>)<\/span>\r\n# \u76ee\u76db\u8a2d\u5b9a<\/span>\r\nax<\/span>.<\/span>set_xticks<\/span>([<\/span>-<\/span>a<\/span>,<\/span> 0<\/span>,<\/span> a<\/span>])<\/span>\r\nax<\/span>.<\/span>set_xticklabels<\/span>(<\/span>labels<\/span> =<\/span> [<\/span>\"$-a$\"<\/span>,<\/span> \"$0$\"<\/span>,<\/span> \"$a$\"<\/span>],<\/span> fontsize<\/span>=<\/span>14<\/span>)<\/span>\r\nax<\/span>.<\/span>set_yticks<\/span>([<\/span>-<\/span>b<\/span>,<\/span> 0<\/span>,<\/span> b<\/span>])<\/span>\r\nax<\/span>.<\/span>set_yticklabels<\/span>(<\/span>labels<\/span> =<\/span> [<\/span>\"$-b$\"<\/span>,<\/span> \"$0$\"<\/span>,<\/span> \"$b$\"<\/span>],<\/span> fontsize<\/span>=<\/span>14<\/span>)<\/span>\r\n\r\n# x \u8ef8\uff0cy \u8ef8<\/span>\r\nax<\/span>.<\/span>axhline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'lightgray'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>)<\/span>\r\nax<\/span>.<\/span>axvline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'lightgray'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ \u306e\u6955\u5186\u306e\u9762\u7a4d\u3002<\/p>\n
$a > b > 0$ \u3068\u3057\u3066 $a$ \u306f\u9577\u534a\u5f84\uff0c$b$ \u306f\u77ed\u534a\u5f84\u3068\u547c\u3076\u3002\u96e2\u5fc3\u7387 $e$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c$b = a \\sqrt{1-e^2}$\u3002<\/p>\n
$\\displaystyle \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ \u3088\u308a $\\displaystyle y = \\pm b \\sqrt{1-\\frac{x^2}{a^2}}$\u3002\u4e0a\u534a\u5206\u306e\u9762\u7a4d\uff0c\u3064\u307e\u308a $\\displaystyle y = b \\sqrt{1-\\frac{x^2}{a^2}}$ \u3068 $x$ \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u30662\u500d\u3059\u308c\u3070\u3088\u3044\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\nS &=& 2 \\int_{-a}^a b \\sqrt{1 \u2013\\frac{x^2}{a^2}} dx \\\\
\n&&\\qquad (x \\equiv a \\sin\\theta, \\ \\sqrt{1 -\\frac{x^2}{a^2}} = \\cos\\theta, \\ dx = a \\cos\\theta d\\theta)\\\\
\n&=& 2 a b \\int_{-\\pi\/2}^{\\pi\/2} \\cos^2\\theta d\\theta\\\\
\n&=& a b \\int_{-\\pi\/2}^{\\pi\/2} (1 + \\cos 2\\theta) d\\theta \\\\
\n&=&a b \\left[ \\theta + \\frac{1}{2}\\sin 2\\theta\\right]_{-\\pi\/2}^{\\pi\/2} \\\\
\n&=& \\pi a b
\n\\end{eqnarray}<\/p>\n
SymPy \u3067\u76f4\u63a5\u7a4d\u5206\u3057\u3066\u307f\u308b\u3068\uff0c\uff08$a>0, b>0$ \u3092\u4eee\u5b9a\u3057\u3066\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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var<\/span>(<\/span>'a b e'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\nvar<\/span>(<\/span>'x y'<\/span>)<\/span>\r\n\r\ndef<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> b<\/span>*<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>\/<\/span>a<\/span>**<\/span>2<\/span>)<\/span>\r\n\r\nEq<\/span>(<\/span>\r\n    2<\/span>*<\/span>Integral<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>a<\/span>,<\/span> a<\/span>)),<\/span>\r\n    2<\/span>*<\/span>integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>a<\/span>,<\/span> a<\/span>))<\/span>\r\n)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
$\\displaystyle 2 \\int\\limits_{-a}^{a} b \\sqrt{1 -\\frac{x^{2}}{a^{2}}}\\, dx = \\pi a b$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u5468<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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a<\/span> =<\/span> 2<\/span>\r\nb<\/span> =<\/span> 1<\/span>\r\np<\/span> =<\/span> plot_geometry<\/span>(<\/span>\r\n    Ellipse<\/span>(<\/span>Point<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>),<\/span> hradius<\/span>=<\/span>a<\/span>,<\/span> vradius<\/span>=<\/span>b<\/span>),<\/span>\r\n    {<\/span>\"color\"<\/span>:<\/span>\"blue\"<\/span>,<\/span> \"lw\"<\/span>:<\/span>2<\/span>},<\/span> is_filled<\/span> =<\/span> False<\/span>,<\/span>\r\n    aspect<\/span> =<\/span> 'equal'<\/span>,<\/span> size<\/span>=<\/span>(<\/span>6.4<\/span>,<\/span>6.4<\/span>),<\/span> n<\/span>=<\/span>500<\/span>,<\/span> show<\/span>=<\/span>False<\/span>);<\/span>\r\n\r\nfig<\/span>,<\/span> ax<\/span> =<\/span> p<\/span>.<\/span>fig<\/span>,<\/span> p<\/span>.<\/span>ax<\/span>\r\nfig<\/span>.<\/span>tight_layout<\/span>()<\/span>\r\n\r\nax<\/span>.<\/span>grid<\/span>(<\/span>linestyle<\/span> =<\/span> 'dotted'<\/span>)<\/span>\r\n# \u8868\u793a\u7bc4\u56f2<\/span>\r\nax<\/span>.<\/span>set_xlim<\/span>(<\/span>-<\/span>a<\/span>-<\/span>0.5<\/span>,<\/span> a<\/span>+<\/span>0.5<\/span>)<\/span>\r\nax<\/span>.<\/span>set_ylim<\/span>(<\/span>-<\/span>b<\/span>-<\/span>0.5<\/span>,<\/span> b<\/span>+<\/span>0.5<\/span>)<\/span>\r\n# \u76ee\u76db\u8a2d\u5b9a<\/span>\r\nax<\/span>.<\/span>set_xticks<\/span>([<\/span>-<\/span>a<\/span>,<\/span> 0<\/span>,<\/span> a<\/span>])<\/span>\r\nax<\/span>.<\/span>set_xticklabels<\/span>(<\/span>labels<\/span> =<\/span> [<\/span>\"$-a$\"<\/span>,<\/span> \"$0$\"<\/span>,<\/span> \"$a$\"<\/span>],<\/span> fontsize<\/span>=<\/span>14<\/span>)<\/span>\r\nax<\/span>.<\/span>set_yticks<\/span>([<\/span>-<\/span>b<\/span>,<\/span> 0<\/span>,<\/span> b<\/span>])<\/span>\r\nax<\/span>.<\/span>set_yticklabels<\/span>(<\/span>labels<\/span> =<\/span> [<\/span>\"$-b$\"<\/span>,<\/span> \"$0$\"<\/span>,<\/span> \"$b$\"<\/span>],<\/span> fontsize<\/span>=<\/span>14<\/span>)<\/span>\r\n\r\n# x \u8ef8\uff0cy \u8ef8<\/span>\r\nax<\/span>.<\/span>axhline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'lightgray'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>)<\/span>\r\nax<\/span>.<\/span>axvline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'lightgray'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u5f0f\u3092\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3059\u308b\u3068\uff0c\uff08$x = a \\cos\\theta, \\ y = b \\sin\\theta$ \u3068\u3059\u308b\u3068\u3053\u308d\u3092\u8da3\u5411\u3092\u5909\u3048\u3066\uff09 $x = a \\sin\\theta, \\ y = b \\cos\\theta$\u3002$0 \\le \\theta \\le \\pi\/2$ \u306e\u9577\u3055\u30924\u500d\u3059\u308c\u3070\u3088\u3044\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\nL &=& 4 \\int_0^{\\frac{\\pi}{2}} \\sqrt{\\left(\\frac{dx}{d\\theta}\\right)^2 + \\left(\\frac{dy}{d\\theta}\\right)^2} \\,d\\theta\\\\
\n&=& 4 \\int_0^{\\frac{\\pi}{2}} \\sqrt{a^2 \\cos^2\\theta + b^2 \\sin^2\\theta} \\,d\\theta \\\\
\n&=& 4 a \\int_0^{\\frac{\\pi}{2}} \\sqrt{\\cos^2\\theta + (1-e^2) \\sin^2\\theta} \\,d\\theta \\\\
\n&=& 4 a \\int_0^{\\frac{\\pi}{2}} \\sqrt{1 -e^2 \\sin^2\\theta} \\,d\\theta \\\\
\n&=& 4 a E(e)
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067
\n$\\displaystyle E(k) \\equiv \\int_0^{\\frac{\\pi}{2}} \\sqrt{1 -k^2 \\sin^2\\theta} \\,d\\theta$ \u306f\u7b2c\u4e8c\u7a2e\u5b8c\u5168\u6955\u5186\u7a4d\u5206\u3067\u3042\u308b\u3002<\/p>\n