{"id":4573,"date":"2023-12-18T10:36:47","date_gmt":"2023-12-18T01:36:47","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=4573"},"modified":"2023-12-18T10:39:34","modified_gmt":"2023-12-18T01:39:34","slug":"maxima-%e3%81%ab%e3%82%88%e3%82%8b%e3%83%91%e3%83%a9%e3%83%91%e3%83%a9%e3%82%a2%e3%83%8b%e3%83%a1%e4%bd%9c%e6%88%90","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/%e5%8f%82%e8%80%83%ef%bc%9amaxima-%e7%b7%a8\/maxima-%e3%81%ab%e3%82%88%e3%82%8b%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/maxima-%e3%81%ab%e3%82%88%e3%82%8b%e3%83%91%e3%83%a9%e3%83%91%e3%83%a9%e3%82%a2%e3%83%8b%e3%83%a1%e4%bd%9c%e6%88%90\/","title":{"rendered":"Maxima \u3067\u30d1\u30e9\u30d1\u30e9\u30a2\u30cb\u30e1\u4f5c\u6210\uff1affmpeg \u3067 mp4 \u7de8"},"content":{"rendered":"

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\u659c\u65b9\u6295\u5c04\u3092\u30d1\u30e9\u30d1\u30e9\u30a2\u30cb\u30e1\u306b\u3059\u308b<\/h3>\n

\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h4>\n

\u6c34\u5e73\u65b9\u5411\u3092 $x$, \u925b\u76f4\u4e0a\u5411\u304d\u3092 $y$ \u3068\u3057\u3066\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u91cd\u529b\u52a0\u901f\u5ea6\u306e\u5927\u304d\u3055\u3092 $g$ \u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\n\\frac{d^2 x}{dt^2} &=& 0 \\\\
\n\\frac{d^2 y}{dt^2} &=& -g
\n\\end{eqnarray}<\/p>\n

\u89e3\u306f<\/p>\n

\\begin{eqnarray}
\nx(t) &=& x_0 + v_{0x} t \\\\
\ny(t) &=& y_0 + v_{0y} t – \\frac{1}{2} g t^2
\n\\end{eqnarray}<\/p>\n

\u521d\u671f\u6761\u4ef6\u3092 $x_0 = 0, y_0 = 0, v_{0x} = v_0 \\cos\\theta, v_{0y} = v_0 \\sin\\theta$ \u3068\u3059\u308b\u3068<\/p>\n

\\begin{eqnarray}
\nx(t) &=& v_{0} \\cos\\theta \\cdot t \\\\
\ny(t) &=& v_{0} \\sin\\theta \\cdot t – \\frac{1}{2} g t^2
\n\\end{eqnarray}<\/p>\n

\u7121\u6b21\u5143\u5316<\/h4>\n

\u3053\u306e\u7cfb\u306b\u7279\u5fb4\u7684\u306a\u6642\u9593 $\\displaystyle \\tau \\equiv \\frac{v_0}{g}$ \u304a\u3088\u3073\u9577\u3055 $\\displaystyle \\ell \\equiv v_0 \\tau = \\frac{v_0^2}{g}$ \u3067\u89e3\u3092\u7121\u6b21\u5143\u5316\u3057\u3066\u304a\u304f\u3002<\/p>\n

\\begin{eqnarray}
\n\\bar{t} &\\equiv& \\frac{t}{\\tau} \\\\
\n\\bar{x} &\\equiv& \\frac{x}{\\ell} = \\cos\\theta\\cdot \\bar{t} \\\\
\n\\bar{y}&\\equiv& \\frac{y}{\\ell} = \\bar{h} + \\sin\\theta\\cdot \\bar{t} – \\frac{1}{2} \\bar{t}^2 \\\\
\n\\bar{v}_x &\\equiv& \\frac{d\\bar{x}}{d\\bar{t}} = \\cos\\theta \\\\
\n\\bar{v}_y &\\equiv& \\frac{d\\bar{y}}{d\\bar{t}} = \\sin\\theta – \\bar{t}
\n\\end{eqnarray}<\/p>\n

\u4ee5\u5f8c\u3057\u3070\u3089\u304f\u306f\uff0c\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u91cf\u3067\u3042\u308b\u3053\u3068\u3092\u5fd8\u308c\u306a\u3044\u3053\u3068\u306b\u3057\u3066\uff0c\u7c21\u5358\u306e\u305f\u3081\u306b $\\bar{\\ }$ \u3092\u7701\u7565\u3059\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n{x} &=& \\cos\\theta\\cdot {t} \\tag{1}\\\\
\n{y} &=&\u00a0 \\sin\\theta\\cdot {t} – \\frac{1}{2} {t}^2 \\tag{2}\\\\
\n{v}_x &=& \\cos\\theta \\tag{3}\\\\
\n{v}_y &=& \\sin\\theta – {t}\u3000 \\tag{4}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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x<\/span>(<\/span>t<\/span>, theta<\/span>)<\/span>:=<\/span> cos<\/span>(<\/span>theta<\/span>)<\/span> *<\/span> t<\/span>$\r\ny<\/span>(<\/span>t<\/span>, theta<\/span>)<\/span>:=<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span> *<\/span> t<\/span> -<\/span> t<\/span>**<\/span>2\/<\/span>2$\r\n\r\nvx<\/span>(<\/span>t<\/span>, theta<\/span>)<\/span>:=<\/span> cos<\/span>(<\/span>theta<\/span>)<\/span>$\r\nvy<\/span>(<\/span>t<\/span>, theta<\/span>)<\/span>:=<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span> -<\/span> t<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6ede\u7a7a\u6642\u9593\u3068\u5230\u9054\u8ddd\u96e2<\/h4>\n
$t = 0$ \u3067\u6295\u5c04\u3057\u3066\uff0c\u3075\u305f\u305f\u3073\u5730\u9762 $y = 0$ \u306b\u843d\u3061\u308b\u307e\u3067\u306e\u6ede\u7a7a\u6642\u9593 $T$ \u306f $(2)$ \u5f0f\u304b\u3089<\/p>\n
$$ 0 = \\sin\\theta \\cdot t – \\frac{1}{2} t^2 $$<\/p>\n
\u3088\u308a\uff0c
\n$$T = 2 \\sin \\theta $$<\/p>\n
\u3053\u306e\u6642\u9593\u3067\u306e\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $L$ \u306f $(1)$ \u5f0f\u304b\u3089<\/p>\n
\\begin{eqnarray}
\nL &=& \\cos\\theta \\cdot T \\\\
\n&=& 2 \\sin\\theta \\cos\\theta = \\sin 2\\theta
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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T<\/span>(<\/span>theta<\/span>)<\/span>:=<\/span> 2<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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draw2d()<\/code> \u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f<\/h4>\n
Maxima \u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u95a2\u6570\u3068\u3057\u3066\u306f plot2d()<\/code>, plot3d()<\/code> \u306e plot<\/code> \u7cfb\u3068\uff0cdraw2d()<\/code> , draw3d()<\/code> \u306e draw<\/code> \u7cfb\u304c\u3042\u308b\u3002\u3053\u3053\u3067\u306f\uff0cdraw<\/code> \u7cfb\u3092\u5229\u7528\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\/* \u03b8 = \u03c0\/4 \u306e\u5834\u5408\u306b\uff0c\u307e\u305a\u306f\u8ecc\u9053\u306e\u5168\u4f53\u3092\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3002*\/<\/span>\r\ndraw2d<\/span>(<\/span>\r\n  \/* \u8868\u793a\u7bc4\u56f2 *\/<\/span>\r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>0.1<\/span>, 1.<\/span>1]<\/span>, yrange<\/span> =<\/span> [<\/span>-<\/span>0.1<\/span>, 0.<\/span>5]<\/span>,\r\n  \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n  proportional_axes<\/span>=<\/span>xy<\/span>,\r\n  \/* \u30b0\u30ea\u30c3\u30c9\u3002y \u8ef8\u306e\u76ee\u76db *\/<\/span>\r\n  grid<\/span> =<\/span> true<\/span>, ytics<\/span> =<\/span> 0.2<\/span>, \r\n  xaxis<\/span> =<\/span> true<\/span>, yaxis<\/span> =<\/span> true<\/span>,\r\n    \r\n  \/* \u5c04\u6cd5\u6295\u5c04\u306e\u8ecc\u9053 *\/<\/span>\r\n  line_width<\/span> =<\/span> 1<\/span>, \r\n  color<\/span> =<\/span> purple<\/span>,\r\n  parametric<\/span>(<\/span>x<\/span>(<\/span>t<\/span>, %pi<\/span>\/<\/span>4<\/span>)<\/span>, y<\/span>(<\/span>t<\/span>, %pi<\/span>\/<\/span>4<\/span>)<\/span>, t<\/span>, 0<\/span>, T<\/span>(<\/span>%pi<\/span>\/<\/span>4<\/span>))<\/span>\r\n  )<\/span>$\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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\u0394t \u3054\u3068\u306e\u30b0\u30e9\u30d5\u3092 N \u500b\u4f5c\u6210<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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\/* \u30d1\u30e9\u30d1\u30e9\u30a2\u30cb\u30e1\u306b\u3059\u308b\u305f\u3081\uff0c\u4e00\u5b9a\u6642\u9593\u9593\u9694\u3054\u3068\u306e\u30b0\u30e9\u30d5\u3092\u4f5c\u6210\u3057\uff0c*\/<\/span>\r\n\/* \u30d5\u30a1\u30a4\u30eb\u306b\u4fdd\u5b58\u3059\u308b *\/<\/span>\r\n\r\n\/* \u5206\u5272\u6570 *\/<\/span>\r\nN<\/span>:<\/span> 40$\r\n\r\nfor<\/span> i<\/span>:<\/span>0<\/span> thru<\/span> N<\/span> do<\/span>(<\/span>\r\n  \/* %03d.png \u306e\u3088\u3046\u306a filename \u306b *\/<\/span>\r\n  tmpname<\/span>:<\/span> printf<\/span>(<\/span>false<\/span>, \"~4d\"<\/span>,1000+<\/span>i<\/span>)<\/span>,\r\n  filename<\/span>:<\/span> substring<\/span>(<\/span>tmpname<\/span>, 2<\/span>)<\/span>, \r\n  \r\n  dt<\/span>:<\/span> T<\/span>(<\/span>%pi<\/span>\/<\/span>4<\/span>)<\/span>\/<\/span>N<\/span>, \r\n  tend<\/span>:<\/span> dt<\/span> *<\/span> i<\/span>, \r\n  \r\n  draw2d<\/span>(<\/span>\r\n    nticks<\/span>=<\/span>1000<\/span>,\r\n    \/* \u8868\u793a\u7bc4\u56f2 *\/<\/span>\r\n    xrange<\/span> =<\/span> [<\/span>-<\/span>0.1<\/span>, 1.<\/span>1]<\/span>, yrange<\/span> =<\/span> [<\/span>-<\/span>0.1<\/span>, 0.<\/span>5]<\/span>,\r\n    \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n    proportional_axes<\/span>=<\/span>xy<\/span>,\r\n    \/* \u30b0\u30ea\u30c3\u30c9\u3002y \u8ef8\u306e\u76ee\u76db *\/<\/span>\r\n    grid<\/span> =<\/span> true<\/span>, ytics<\/span> =<\/span> 0.2<\/span>, \r\n    xaxis<\/span> =<\/span> true<\/span>, yaxis<\/span> =<\/span> true<\/span>,\r\n    \r\n    \/* \u659c\u65b9\u6295\u5c04\u306e\u8ecc\u9053 *\/<\/span>\r\n    line_width<\/span> =<\/span> 1<\/span>, \r\n    color<\/span> =<\/span> purple<\/span>,\r\n    parametric<\/span>(<\/span>x<\/span>(<\/span>t<\/span>, %pi<\/span>\/<\/span>4<\/span>)<\/span>, y<\/span>(<\/span>t<\/span>, %pi<\/span>\/<\/span>4<\/span>)<\/span>, t<\/span>, 0<\/span>, tend<\/span>)<\/span>,\r\n    \r\n    \/* \u5404 i \u3054\u3068\u306b png \u30d5\u30a1\u30a4\u30eb\u306b\u4fdd\u5b58\u3059\u308b\u3002*\/<\/span> \r\n    file_name<\/span> =<\/span> filename<\/span>, \r\n    dimensions<\/span> =<\/span> 60*<\/span>[<\/span>12.0<\/span>,9.<\/span>0]<\/span>,\r\n    terminal<\/span> =<\/span> '<\/span>png<\/span> \r\n  )<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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ffmpeg \u3067\u30a2\u30cb\u30e1\u306b\u3057\u3066\u307f\u308b<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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\/* \u53e4\u3044 out.mp4 \u306f\u524a\u9664\u3057\u3066\u304b\u3089\u3002*\/<\/span>\r\nsystem<\/span>(<\/span>\"rm -f out.mp4\"<\/span>)<\/span>$\r\n\r\n\/* framerate \u3092\u5909\u3048\u308b\u3068\u30d1\u30e9\u30d1\u30e9\u30a2\u30cb\u30e1\u306e\u901f\u3055\u304c\u304b\u308f\u308b\u3002*\/<\/span>\r\n\/* stream_loop \u306f\u3055\u3089\u306b\u4f55\u56de\u7e70\u308a\u8fd4\u3059\u304b\u3092\u6307\u5b9a\u3059\u308b\u3002*\/<\/span>\r\nsystem<\/span>(<\/span>\"ffmpeg  -stream_loop 2  -framerate 10    -hide_banner -loglevel error -i %03d.png -vcodec libx264 -pix_fmt yuv420p out.mp4\"<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3067\u304d\u305f\u52d5\u753b\u30d5\u30a1\u30a4\u30eb out.mp4<\/code> \u306f JupyterHub \u306e\u30db\u30fc\u30e0\u304b\u3089\uff0c\u5f53\u8a72\u30d5\u30a1\u30a4\u30eb\u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u3053\u3068\u3067\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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