{"id":4081,"date":"2022-10-24T17:53:30","date_gmt":"2022-10-24T08:53:30","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4081"},"modified":"2023-03-14T16:58:37","modified_gmt":"2023-03-14T07:58:37","slug":"maxima-%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\/4081\/","title":{"rendered":"Maxima \u3067\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3068\u6570\u5024\u89e3"},"content":{"rendered":"

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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 Maxima \u306e\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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Ubes<\/span>(<\/span>N<\/span>, e<\/span>, omegat<\/span>)<\/span>:=<\/span> \r\n  omegat<\/span> +<\/span> sum<\/span>(<\/span>2\/<\/span>n<\/span> *<\/span> bessel_j<\/span>(<\/span>n<\/span>, n<\/span>*<\/span>e<\/span>)<\/span> *<\/span> sin<\/span>(<\/span>n<\/span> *<\/span> omegat<\/span>)<\/span>, n<\/span>, 1<\/span>, N<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}{\\it Ubes}\\left(N , e , {\\it omegat}\\right):={\\it omegat}+{\\it sum}\\left(\\frac{2}{n}\\,{\\it bessel\\_j}\\left(n , n\\,e\\right)\\,\\sin \\left(n\\,{\\it omegat}\\right) , n , 1 , N\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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\/* \u305f\u3068\u3048\u3070 N=10, e=0.5, omegat = 1.0 \u306e\u5024 *\/<\/span>\r\nUbes<\/span>(<\/span>10<\/span>, 0.5<\/span>, 1.0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}1.49885975062147\\]<\/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 $u(n, e, \\omega t)$ \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[3]:<\/div>\n
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Uite<\/span>(<\/span>N<\/span>, e<\/span>, omegat<\/span>)<\/span>:=<\/span>\r\nblock<\/span>(<\/span>  \r\n  if<\/span> N<\/span> =<\/span> 0<\/span> then<\/span>\r\n    omegat<\/span>\r\n  else<\/span>\r\n    omegat<\/span> +<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>Uite<\/span>(<\/span>N<\/span>-<\/span>1<\/span>, e<\/span>, omegat<\/span>))<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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\/* \u305f\u3068\u3048\u3070 N=10, e=0.5, omegat = 1.0 \u306e\u5024 *\/<\/span>\r\nUite<\/span>(<\/span>10<\/span>, 0.5<\/span>, 1.<\/span>0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}1.498701133517836\\]<\/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\uff0cfind_root()<\/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[5]:<\/div>\n
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Unum<\/span>(<\/span>e<\/span>, omegat<\/span>)<\/span>:=<\/span> \r\n  block<\/span>(<\/span>find_root<\/span>(<\/span>u<\/span> -<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span> =<\/span> omegat<\/span>, u<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}{\\it Unum}\\left(e , {\\it omegat}\\right):=\\mathbf{block}\\;\\left({\\it find\\_root}\\left(u-e\\,\\sin u={\\it omegat} , u , 0 , 2\\,\\pi\\right)\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/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>, 1.0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}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[7]:<\/div>\n
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omt<\/span>:<\/span> makelist<\/span>(<\/span>float<\/span>(<\/span>%pi<\/span>\/<\/span>10*<\/span>i<\/span>)<\/span>, i<\/span>, 1<\/span>, 9<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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\/* \u6570\u5024\u89e3 Unum \u3068\u9010\u6b21\u8fd1\u4f3c\u89e3 Uite, \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u89e3 Ubes \u306e\u5dee *\/<\/span>\r\ne<\/span>:<\/span> 0.<\/span>5$\r\nN<\/span>:<\/span> 10$\r\nprintf<\/span>(<\/span>true<\/span>,\" omega t    Unum               Uite-Unum          Ubes-Unum ~%\"<\/span>)<\/span>$\r\n\r\nfor<\/span> i<\/span> thru<\/span> length<\/span>(<\/span>omt<\/span>)<\/span> do<\/span>(<\/span>\r\n  printf<\/span>(<\/span>true<\/span>, \"~8,5f ~18,15f ~18,15f ~18,15f~%\"<\/span>, \r\n                omt<\/span>[<\/span>i<\/span>]<\/span>,\r\n                Unum<\/span>(<\/span>e<\/span>, omt<\/span>[<\/span>i<\/span>])<\/span>, \r\n                Uite<\/span>(<\/span>N<\/span>, e<\/span>, omt<\/span>[<\/span>i<\/span>])<\/span>-<\/span>Unum<\/span>(<\/span>e<\/span>, omt<\/span>[<\/span>i<\/span>])<\/span>, \r\n                Ubes<\/span>(<\/span>N<\/span>, e<\/span>, omt<\/span>[<\/span>i<\/span>])<\/span>-<\/span>Unum<\/span>(<\/span>e<\/span>, omt<\/span>[<\/span>i<\/span>]))<\/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  0.000205334548063\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 -0.000043154412396\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