{"id":2486,"date":"2022-03-15T15:04:51","date_gmt":"2022-03-15T06:04:51","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=2486"},"modified":"2023-03-14T16:55:12","modified_gmt":"2023-03-14T07:55:12","slug":"%e9%80%90%e6%ac%a1%e8%bf%91%e4%bc%bc%e6%b3%95%e3%81%ab%e3%82%88%e3%82%8b%e3%82%b1%e3%83%97%e3%83%a9%e3%83%bc%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e7%b4%a0%e6%9c%b4%e3%81%aa%e8%bf%91%e4%bc%bc%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/2486\/","title":{"rendered":"\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u7d20\u6734\u306a\u8fd1\u4f3c\u89e3\u6cd5"},"content":{"rendered":"

\u540d\u8457\u300c\u5929\u4f53\u3068\u8ecc\u9053\u306e\u529b\u5b66\u300d\uff08\u54c1\u5207\u30fb\u91cd\u7248\u672a\u5b9a<\/a>\u3067 Amazon \u3067\u306f\u304b\u306a\u308a\u306e\u9ad8\u5024<\/a>\uff09\u306e\u300c2.7 \u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u300d\u306b\u306f\u300c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u306b\u306f\u5b9f\u306b\u3055\u307e\u3056\u307e\u306a\u65b9\u6cd5\u304c\u3042\u308b\u300d\u3068\u66f8\u3044\u3066\u3044\u308b\u3002\u3053\u3053\u3067\u306f\uff0cMaxima-Jupyter \u3067\u6570\u5024\u7684\u306b\u89e3\u304f\u65b9\u6cd5\uff08\u5225\u4ef6\u3067\u65e2\u306b\u7d39\u4ecb\u6e08\u307f<\/a>\uff09\u3068\uff0c\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u6975\u3081\u3066\u7d20\u6734\u306a\u8fd1\u4f3c\u89e3\u6cd5\u306b\u3064\u3044\u3066\uff0cMaxima \u306b\u304a\u3051\u308b\u95a2\u6570\u306e\u518d\u5e30\u7684\u5b9a\u7fa9<\/strong><\/span>\u306e\u7df4\u7fd2\u554f\u984c\u3082\u304b\u306d\u3066\u30e1\u30e2\u3057\u3066\u304a\u304f\u3002<\/p>\n

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\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f<\/h3>\n

\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f<\/p>\n

$$ u – e \\sin u = \\frac{2\\pi}{T} t \\equiv \\omega t$$<\/p>\n

\u306b\u3064\u3044\u3066\uff0c$u$ \u3092 $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u8fd1\u4f3c\u7684\u306b\u8868\u3059\u3068\u3044\u3046\u8a71\u3002<\/p>\n

Maxima \u306b\u304a\u3051\u308b\u95a2\u6570\u306e\u518d\u5e30\u7684\u5b9a\u7fa9<\/strong><\/span>\u306e\u7df4\u7fd2\u554f\u984c\u3068\u3057\u3066\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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<\/div>\n
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find_root()<\/code> \u95a2\u6570\u306b\u3088\u308b\u6570\u5024\u7684\u89e3\u6cd5<\/h3>\n

\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306f\u8d85\u8d8a\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u305f\u3081\uff0c$u$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066 $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002\u305d\u3053\u3067\uff0cMaxima \u306e find_root()<\/code> \u95a2\u6570\u3092\u4f7f\u3063\u3066\u6570\u5024\u7684\u306b\u89e3\u304f\u3002<\/p>\n

\u5468\u671f $T$ \u3092 $N$ \u7b49\u5206\u3057\uff0c
\n$$t_i = \\frac{T}{N} \\times i, \\quad (i = 0, 1, \\dots, N)$$<\/p>\n

\u306b\u5bfe\u3057\u3066\uff0c$u_i = u(t_i)$ \u3092\u6570\u5024\u7684\u306b\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
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In\u00a0[1]:<\/div>\n
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\/* \u96e2\u5fc3\u7387 *\/<\/span>\r\ne<\/span>:<\/span> 6\/<\/span>10$\r\n\r\n\/* \u5206\u5272\u6570 *\/<\/span>\r\nN<\/span>:<\/span> 36$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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$$g(u_i) \\equiv u_i – e \\sin u_i = \\frac{2\\pi}{T} t_i = \\frac{2\\pi}{N} \\times i$$\u3092 $u_i$ \u306b\u3064\u3044\u3066 find_root()<\/code> \u95a2\u6570\u3067\u6570\u5024\u7684\u306b\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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g<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> u<\/span> -<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}g\\left(u\\right):=u-e\\,\\sin u\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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for<\/span> i<\/span>:<\/span>0<\/span> thru<\/span> N<\/span> do<\/span> \r\n    u<\/span>[<\/span>i<\/span>]<\/span>:<\/span> find_root<\/span>(<\/span>g<\/span>(<\/span>u<\/span>)<\/span> =<\/span> 2*<\/span>%pi<\/span>\/<\/span>N<\/span> *<\/span> i<\/span>, u<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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find_root()<\/code> \u95a2\u6570\u3067\u6c42\u3081\u305f u[i]<\/code> \u3092\u3042\u3089\u305f\u3081\u3066 g(u)<\/code> \u306b\u4ee3\u5165\u3057\u3066\uff0c\u8aa4\u5dee<\/p>\n
$$\\left|g(u_i) – \\frac{2\\pi}{N} \\times i\\right|$$<\/p>\n
\u3092\u8868\u793a\u3059\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8aa4\u5dee\u306f $10^{-15}$ \u3088\u308a\u5c0f\u3055\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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err<\/span>:<\/span> makelist<\/span>(<\/span>abs<\/span>(<\/span>g<\/span>(<\/span>u<\/span>[<\/span>i<\/span>])<\/span> -<\/span> float<\/span>(<\/span>2*<\/span>%pi<\/span>\/<\/span>N<\/span> *<\/span> i<\/span>))<\/span>, i<\/span>, 0<\/span>, N<\/span>)<\/span>$\r\nlmax<\/span>(<\/span>err<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}8.881784197001252 \\times 10^{-16}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u7d20\u6734\u306a\u8fd1\u4f3c\u7684\u89e3\u6cd5<\/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} \\\\
\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[5]:<\/div>\n
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U<\/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>U<\/span>(<\/span>n<\/span>-<\/span>1<\/span>, e<\/span>, omegat<\/span>))<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u3053\u306e\u5b9a\u7fa9\u304b\u3089\u63a8\u6e2c\u3055\u308c\u308b\u3088\u3046\u306b\uff0c$U(n, e, \\omega t)$ \u306f $e^n$ \u7a0b\u5ea6\u306e\u7cbe\u5ea6\u3068\u8003\u3048\u3089\u308c\u308b\u306e\u3067\uff0c\u4eca\u56de\u306e\u4f8b\u306e\u3088\u3046\u306b $e = 0.6$ \u3060\u3068\uff0c\u4f8b\u3048\u3070\u8aa4\u5dee\u3092 $10^{-15}$ \u3088\u308a\u5c0f\u3055\u304f\u3057\u3088\u3046\u306a\u3069\u3068\u8003\u3048\u308b\u3068\uff0c$n$ \u306e\u5024\u3092\u304b\u306a\u308a\u5927\u304d\u304f\u3057\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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e<\/span>;\r\nfloat<\/span>(<\/span>e<\/span>**<\/span>68<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}\\frac{3}{5}\\]<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}8.208901151521337 \\times 10^{-16}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u4ee5\u4e0b\u3067\u306f\u4f8b\u3068\u3057\u3066 $n = 70$ \u3068\u3057\u3066\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u8fd1\u4f3c\u89e3 $U(70, e, \\omega t)$ \u3068\u6570\u5024\u89e3 u[i]<\/code> \u3068\u306e\u8aa4\u5dee\u306e\u6700\u5927\u5024\u3092\u8868\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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err<\/span>:<\/span> makelist<\/span>(<\/span>abs<\/span>(<\/span>u<\/span>[<\/span>i<\/span>]<\/span> -<\/span> U<\/span>(<\/span>70<\/span>, e<\/span>, float<\/span>(<\/span>2*<\/span>%pi<\/span>\/<\/span>N<\/span> *<\/span> i<\/span>)))<\/span>, i<\/span>, 1<\/span>, N<\/span>)<\/span>$\r\nlmax<\/span>(<\/span>err<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}8.881784197001252 \\times 10^{-16}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u540d\u8457\u300c\u5929\u4f53\u3068\u8ecc\u9053\u306e\u529b\u5b66\u300d\uff08\u54c1\u5207\u30fb\u91cd\u7248\u672a\u5b9a\u3067 Amazon \u3067\u306f\u304b\u306a\u308a\u306e\u9ad8\u5024\uff09\u306e\u300c2.7 \u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u300d\u306b\u306f\u300c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u306b\u306f\u5b9f\u306b\u3055\u307e\u3056\u307e\u306a\u65b9\u6cd5\u304c\u3042\u308b\u300d\u3068\u66f8\u3044\u3066\u3044\u308b\u3002\u3053\u3053\u3067\u306f\uff0cMaxima-Jupyter \u3067\u6570\u5024\u7684\u306b\u89e3\u304f\u65b9\u6cd5\uff08\u5225\u4ef6\u3067\u65e2\u306b\u7d39\u4ecb\u6e08\u307f\uff09\u3068\uff0c\u9010\u6b21\u8fd1\u4f3c\u6cd5\u306b\u3088\u308b\u6975\u3081\u3066\u7d20\u6734\u306a\u8fd1\u4f3c\u89e3\u6cd5\u306b\u3064\u3044\u3066\uff0cMaxima \u306b\u304a\u3051\u308b\u95a2\u6570\u306e\u518d\u5e30\u7684\u5b9a\u7fa9\u306e\u7df4\u7fd2\u554f\u984c\u3082\u304b\u306d\u3066\u30e1\u30e2\u3057\u3066\u304a\u304f\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[14,18],"tags":[],"class_list":["post-2486","post","type-post","status-publish","format-standard","hentry","category-maxima","category-18","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/2486","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2486"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/2486\/revisions"}],"predecessor-version":[{"id":2892,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/2486\/revisions\/2892"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2486"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=2486"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=2486"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}