{"id":4087,"date":"2022-10-24T18:16:50","date_gmt":"2022-10-24T09:16:50","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4087"},"modified":"2023-03-14T16:58:24","modified_gmt":"2023-03-14T07:58:24","slug":"python-%e3%81%ae-sympy-%e3%81%a7%e3%82%b1%e3%83%97%e3%83%a9%e3%83%bc%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e8%bf%91%e4%bc%bc%e8%a7%a3%e3%81%a8%e6%95%b0%e5%80%a4%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4087\/","title":{"rendered":"Python \u306e SymPy \u3067\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3068\u6570\u5024\u89e3"},"content":{"rendered":"

\u300cMaxima \u3067\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3068\u6570\u5024\u89e3<\/a>\u300d\u306e SymPy \u7248\u3002<\/p>\n

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In\u00a0[1]:<\/div>\n
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from<\/span> sympy<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> pi<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u306e\u5927\u304d\u3055\u306e\u8a2d\u5b9a\uff0cinline svg \u8868\u793a<\/span>\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'figure.figsize'<\/span>]<\/span> =<\/span>8<\/span>,<\/span>6<\/span>\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'font.size'<\/span>]<\/span> =<\/span> 12<\/span>\r\nfrom<\/span> IPython.display<\/span> import<\/span> set_matplotlib_formats<\/span>\r\n%<\/span>matplotlib<\/span> inline\r\nset_matplotlib_formats<\/span>(<\/span>'svg'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30c3\u30bb\u30eb\u95a2\u6570\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3<\/h3>\n
\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f<\/p>\n
$$u – e \\sin u = \\omega t$$<\/p>\n
\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3\u306f\uff0c\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 $J_n(x)$ \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n
$$u = \\omega t+ \\sum_{n=1}^{\\infty} \\frac{2}{n} J_n(n e) \\sin(n\\, \\omega t)$$<\/p>\n
\u73fe\u5b9f\u4e16\u754c\u3067\u306f\u7121\u9650\u5927\u307e\u3067\u8db3\u3057\u7b97\u306f\u3067\u304d\u306a\u3044\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b $N$ \u307e\u3067\u306e\u548c\u3092\u3068\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n
$$u(N, e, \\omega t) = \\omega t+ \\sum_{n=1}^{N} \\frac{2}{n} J_n(n e) \\sin(n\\, \\omega t)$$<\/p>\n
\u3053\u308c\u3092\u95a2\u6570 Ubes(N, e, omegat)<\/code> \u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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omegat<\/span> =<\/span> Symbol<\/span>(<\/span>'omegat'<\/span>)<\/span>\r\n\r\ndef<\/span> Ubes<\/span>(<\/span>N<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>):<\/span>\r\n    return<\/span>(<\/span>omegat<\/span> +<\/span> \r\n           summation<\/span>(<\/span>2<\/span>\/<\/span>n<\/span> *<\/span> besselj<\/span>(<\/span>n<\/span>,<\/span> n<\/span>*<\/span>e<\/span>)<\/span> *<\/span> sin<\/span>(<\/span>n<\/span> *<\/span> omegat<\/span>),<\/span> (<\/span>n<\/span>,<\/span> 1<\/span>,<\/span> N<\/span>))<\/span>\r\n          )<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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Ubes<\/span>(<\/span>2<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
$\\displaystyle omegat + 2 \\sin{\\left(omegat \\right)} J_{1}\\left(e\\right) + \\sin{\\left(2 omegat \\right)} J_{2}\\left(2 e\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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# \u305f\u3068\u3048\u3070 N=10, e=0.5, omegat = 1.0 \u306e\u5024 <\/span>\r\nprint<\/span>(<\/span>float<\/span>(<\/span>Ubes<\/span>(<\/span>10<\/span>,<\/span> 0.5<\/span>,<\/span> 1.0<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1.49885975062147\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3<\/h3>\n
\u96e2\u5fc3\u7387 $e$ \u306f $0 \\leq e < 1$ \u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f
\n$$u – e \\sin u = \\omega t$$
\n\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u9010\u6b21\u8fd1\u4f3c\u7684\u306b\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\nu &=& \\omega t + e \\sin u \\\\
\nu_0 &=& \\omega t\\\\
\nu_1 &=& \\omega t + e \\sin u_0 = \\omega t + e \\sin \\omega t\\\\
\nu_2 &=& \\omega t + e \\sin u_1 =\\omega t + e \\sin\\left(\\omega t + e \\sin \\omega t\\right) \\\\
\nu_3 &=& \\omega t + e \\sin u_2 =\\omega t + e \\sin\\left\\{\\omega t + e \\sin\\left(\\omega t + e \\sin \\omega t\\right)\\right\\} \\\\
\n&\\vdots&\\\\
\nu_{n} &=& \\omega t + e \\sin u_{n-1} = \\dots\\\\
\n\\end{eqnarray}<\/p>\n
$n$ \u304c\u5927\u304d\u304f\u306a\u308b\u3068\uff0c\u5165\u308c\u5b50\u306b\u306a\u3063\u3066\u3044\u308b\u9805\u304c\u3069\u3093\u3069\u3093\u5897\u6b96\u3057\u3066\u3044\u304d\u307e\u3059\u304c\uff0c$u_3$ \u306e\u3042\u305f\u308a\u307e\u3067\u306f\uff0c\u8fd1\u4f3c\u7684\u306b $u$ \u306f $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u3042\u3089\u308f\u3055\u308c\u3066\u3044\u308b\u306a\u3041… \u3068\u3044\u3046\u898b\u305f\u76ee\u304c\u3057\u307e\u3059\u3002<\/p>\n
\u4e0a\u306e\u5f0f\u306b\u305d\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u95a2\u6570 Uite(N, e, omegat)<\/code> \u3092\u518d\u5e30\u7684\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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def<\/span> Uite<\/span>(<\/span>N<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>):<\/span>\r\n    if<\/span> N<\/span> ==<\/span>0<\/span>:<\/span>\r\n        return<\/span>(<\/span>omegat<\/span>)<\/span>\r\n    else<\/span>:<\/span>\r\n        return<\/span>(<\/span>omegat<\/span> +<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>Uite<\/span>(<\/span>N<\/span>-<\/span>1<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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Uite<\/span>(<\/span>3<\/span>,<\/span> e<\/span>,<\/span> omegat<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle e \\sin{\\left(e \\sin{\\left(e \\sin{\\left(omegat \\right)} + omegat \\right)} + omegat \\right)} + omegat$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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# \u305f\u3068\u3048\u3070 N=10, e=0.5, omegat = 1.0 \u306e\u5024<\/span>\r\nprint<\/span>(<\/span>float<\/span>(<\/span>Uite<\/span>(<\/span>10<\/span>,<\/span> 0.5<\/span>,<\/span> 1.0<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1.4987011335178357\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3<\/h3>\n
$\\omega t$ \u3092\u4e0e\u3048\u305f\u3068\u304d\u306b\uff0c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066 $u(\\omega t)$ \u3092\u6c42\u3081\u308b\u3002\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306f\u8d85\u8d8a\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u306e\u3067\uff0cnsolve()<\/code> \u95a2\u6570\u3092\u4f7f\u3063\u3066\u6570\u5024\u306b\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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def<\/span> Unum<\/span>(<\/span>e<\/span>,<\/span> omegat<\/span>):<\/span>\r\n    return<\/span>(<\/span>nsolve<\/span>(<\/span>u<\/span>-<\/span>e<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>-<\/span>omegat<\/span>,<\/span> u<\/span>,<\/span> omegat<\/span>,<\/span> prec<\/span>=<\/span>16<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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# \u305f\u3068\u3048\u3070 e=0.5, omegat = 1.0 \u306e\u5024<\/span>\r\nUnum<\/span>(<\/span>0.5<\/span>,<\/span> 1.0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle 1.498701133517848$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u89e3\u3068\u8fd1\u4f3c\u89e3\u3068\u306e\u6bd4\u8f03<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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# \u6570\u5024\u89e3 Unum \u3068\u9010\u6b21\u8fd1\u4f3c\u89e3 Uite, \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3 Ubes \u306e\u5dee<\/span>\r\ne<\/span> =<\/span> 0.5<\/span>\r\nN<\/span> =<\/span> 10<\/span>\r\n\r\nprint<\/span>(<\/span>\" omega t    Unum               Uite-Unum          Ubes-Unum \"<\/span>)<\/span>\r\nfor<\/span> i<\/span> in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> 10<\/span>):<\/span>\r\n    omti<\/span> =<\/span> float<\/span>(<\/span>pi<\/span>\/<\/span>10<\/span>*<\/span>i<\/span>)<\/span>\r\n    print<\/span>(<\/span>'<\/span>%8.5f<\/span> %18.15f<\/span> %18.15f<\/span> %18.15f<\/span>'<\/span> %<\/span> \r\n          (<\/span>omti<\/span>,<\/span> \r\n           Unum<\/span>(<\/span>e<\/span>,<\/span> omti<\/span>),<\/span> \r\n           Uite<\/span>(<\/span>N<\/span>,<\/span> e<\/span>,<\/span> omti<\/span>)<\/span> -<\/span> Unum<\/span>(<\/span>e<\/span>,<\/span> omti<\/span>),<\/span> \r\n           Ubes<\/span>(<\/span>N<\/span>,<\/span> e<\/span>,<\/span> omti<\/span>)<\/span> -<\/span> Unum<\/span>(<\/span>e<\/span>,<\/span> omti<\/span>)<\/span>\r\n          )<\/span>\r\n         )<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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 omega t    Unum               Uite-Unum          Ubes-Unum \r\n 0.31416  0.593999023813608 -0.000048370871788  0.000205334548064\r\n 0.62832  1.065940683889791 -0.000000479808530 -0.000233511559215\r\n 0.94248  1.438080909968085 -0.000000000003061  0.000199446542732\r\n 1.25664  1.748741781633489  0.000000000005287 -0.000159160439054\r\n 1.57080  2.020979938089770 -0.000000056633642  0.000123562038969\r\n 1.88496  2.268208852924498 -0.000003478720316 -0.000093184551055\r\n 2.19911  2.498822425235399 -0.000028291765788  0.000066804145119\r\n 2.51327  2.718544855625697 -0.000076315203530 -0.000043154412395\r\n 2.82743  2.931640124182721 -0.000080693322920  0.000021180282106\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6570\u5024\u89e3\u3068\u306e\u5dee\u306e\u4e8c\u4e57\u5e73\u5747\u5e73\u65b9\u6839<\/h3>\n