{"id":230,"date":"2022-01-05T15:31:43","date_gmt":"2022-01-05T06:31:43","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=230"},"modified":"2024-08-06T16:53:25","modified_gmt":"2024-08-06T07:53:25","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e5%9b%9e%e8%bb%a2%e7%b3%bb%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e9%80%9f%e5%ba%a6%e3%83%bb%e5%8a%a0%e9%80%9f%e5%ba%a6%e3%83%99%e3%82%af%e3%83%88%e3%83%ab","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e6%99%82%e7%a9%ba%e3%81%ae%e8%a1%a8%e3%81%97%e6%96%b9\/3%e6%ac%a1%e5%85%83%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e5%be%ae%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9a%e5%9b%9e%e8%bb%a2%e7%b3%bb%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e9%80%9f%e5%ba%a6%e3%83%bb%e5%8a%a0%e9%80%9f%e5%ba%a6%e3%83%99%e3%82%af%e3%83%88%e3%83%ab\/","title":{"rendered":"\u88dc\u8db3\uff1a\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u901f\u5ea6\u30fb\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb"},"content":{"rendered":"

<\/p>\n

\u9759\u6b62\u6163\u6027\u7cfb\u306b\u304a\u3051\u308b\u901f\u5ea6\u30fb\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/h3>\n

\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u306e\u6642\u9593\u5fae\u5206\u306e\u7d50\u679c\u3092\u7528\u3044\u3066\uff0c\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\uff0c\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u306b\u3064\u3044\u3066\u304a\u3055\u3089\u3044\u3059\u308b\u3002<\/p>\n

\u9759\u6b62\u6163\u6027\u7cfb\u306b\u304a\u3044\u3066\u306f\uff0c\u901f\u5ea6\u30d9\u30af\u30c8\u30eb \\(\\vec{v}\\) \u3068\u306f\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb \\(\\vec{r}\\) \u306e\u6642\u9593\u5fae\u5206\u3067\u3042\u308b\u3068\u7fd2\u3063\u305f\u304b\u3082\u3057\u308c\u306a\u3044\u3002
\n$$\\vec{v} = \\frac{d\\vec{r}}{dt} = \\dot{\\vec{r}}$$\u540c\u69d8\u306b\uff0c\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb \\(\\vec{a}\\) \u306f\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u6642\u9593\u5fae\u5206\u3068\u3057\u3066
\n$$\\vec{a} = \\frac{d\\vec{v}}{dt} = \\dot{\\vec{v}} = \\frac{d^2\\vec{r}}{dt^2} = \\ddot{\\vec{r}}$$<\/p>\n

\u3057\u304b\u3057\uff0c\u3053\u308c\u306f\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u304c\u6642\u9593\u7684\u306b\u5909\u5316\u3057\u306a\u3044\u3068\u3044\u3046\u6761\u4ef6\u306e\u3082\u3068\u3067\u306e\u8a71\u3067\u3042\u308b\u3002\u56de\u8ee2\u7cfb\u306e\u3088\u3046\u306b\uff0c\u4e00\u822c\u306b\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\u3067\u306a\u3044\u5834\u5408\u306b\u306f\uff0c\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u3068\u306f\u4f55\u304b\uff0c\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u3068\u306f\u4f55\u304b\u3092\u306f\u3063\u304d\u308a\u3055\u305b\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n

\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u30fb\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u7fa9<\/h3>\n

\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\u3067\u306a\u3044\u3088\u3046\u306a\u4e00\u822c\u7684\u72b6\u6cc1\u306e\u3082\u3068\u3067\uff0c\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3068\u306f\uff0c\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306e\u6642\u9593\u5fae\u5206\u3092\u6210\u5206\u3068\u3059\u308b\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3064\u307e\u308a\uff0c\\(S\\) \u7cfb\u306b\u304a\u3051\u308b\u901f\u5ea6\u30d9\u30af\u30c8\u30eb \\(\\vec{v} \\) \u306f<\/p>\n

$$\\vec{v} \\equiv \\dot{x}\\, \\vec{e}_x + \\dot{y} \\,\\vec{e}_y +\\dot{z}\\, \\vec{e}_z$$ \u3067\u3042\u308a\uff0c\\(S’\\) \u7cfb\u306b\u304a\u3051\u308b\u901f\u5ea6\u30d9\u30af\u30c8\u30eb \\(\\vec{v}’ \\) \u306f<\/p>\n

$$\\vec{v}’ \\equiv \\dot{x}’ \\vec{e}’_x + \\dot{y}’ \\vec{e}’_y +\\dot{z}’ \\vec{e}’_z$$ \u3068\u306a\u308b\u3002\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\u3067\u306a\u3044\u5834\u5408\u306b\u306f\u4e00\u822c\u306b\uff0c
\n$$\\vec{v}’ \\neq \\frac{{\\ }d\\vec{r}’}{dt}$$ \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n

\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u3082\uff0c\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u6210\u5206\u306e\u5fae\u5206\u3092\u6210\u5206\u3068\u3059\u308b\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c\\(S\\) \u7cfb\u306b\u304a\u3051\u308b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb \\(\\vec{a} \\) \u306f
\n\\begin{eqnarray}
\n\\vec{a} &\\equiv& \\ddot{x}\\,\\vec{e}_x + \\ddot{y}\\,\\vec{e}_y + \\ddot{z}\\,\\vec{e}_z
\n\\end{eqnarray}\u3067\u3042\u308a\uff0c\\(S’\\) \u7cfb\u306b\u304a\u3051\u308b\u52a0\u901f\u5ea6\u30d9\u30af\u30c8\u30eb \\(\\vec{a}’ \\) \u306f<\/p>\n

\\begin{eqnarray}
\n\\vec{a}’ &\\equiv& \\ddot{x}’\\,\\vec{e}’_x + \\ddot{y}’\\,\\vec{e}’_y + \\ddot{z}’\\,\\vec{e}’_z
\n\\end{eqnarray} \u3068\u306a\u308b\u3002\u57fa\u672c\u30d9\u30af\u30c8\u30eb\u304c\u4e00\u5b9a\u3067\u306a\u3044\u5834\u5408\u306b\u306f\u4e00\u822c\u306b\uff0c
\n$$\\vec{a}’ \\neq \\frac{{\\ }d\\vec{v}’}{dt}$$ \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n

\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u6642\u9593\u5fae\u5206<\/h3>\n

\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u6642\u9593\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\uff0c\\(\\vec{v}\\) \u3068 \\(\\vec{v}’\\) \u3068\u306e\u95a2\u4fc2\u3084\uff0c\\( \\vec{a}\\) \u3068 \\(\\vec{a}’\\) \u3068\u306e\u95a2\u4fc2\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002
\n\\begin{eqnarray}
\n\\vec{v} = \\dot{\\vec{r}} = \\dot{\\vec{r}}’ &=& (x’\\,\\vec{e}’_{x} + y’\\,\\vec{e}’_{y} + z’\\,\\vec{e}’_{z})\\dot{} \\\\
\n&=& (\\dot{x}’ \\vec{e}’_{x} + \\dot{y}’ \\vec{e}’_{y} + \\dot{x}’ \\vec{e}’_{x}) \\\\
\n&&\\quad + (x’\\,\\dot{\\vec{e}}’_{x} + y’\\,\\dot{\\vec{e}}’_{y} + z’\\,\\dot{\\vec{e}}’_{z}) \\\\
\n&=& \\vec{v}’ + \\vec{\\omega}\\times(x’\\,\\vec{e}’_{x} + y’\\,\\vec{e}’_{y} + z’\\,\\vec{e}’_{z}) \\\\
\n&=& \\vec{v}’ + \\vec{\\omega}\\times\\vec{r}’
\n\\end{eqnarray}
\n$$\\therefore\\ \\ \\vec{v} = \\vec{v}’ + \\vec{\\omega}\\times\\vec{r}’ $$<\/p>\n

\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\u89d2\u901f\u5ea6\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\vec{\\omega}\\) \u304c\u4e00\u5b9a\u3067\u3042\u308b\u3068\u3057\u3066\u8a08\u7b97\u3092\u7d9a\u3051\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\vec{a} = \\ddot{\\vec{r}} = \\ddot{\\vec{r}}’ &=&\u00a0 \\dot{\\vec{v}}’ + \\vec{\\omega}\\times\\dot{\\vec{r}}’ \\\\
\n&=& (\\vec{a}’ + \\vec{\\omega}\\times\\vec{v}’) + \\vec{\\omega}\\times \\left(\\vec{v}’ + \\vec{\\omega}\\times\\vec{r}’\\right) \\\\
\n&=& \\vec{a}’ + 2 \\vec{\\omega}\\times\\vec{v}’ + \\vec{\\omega}\\times(\\vec{\\omega}\\times\\vec{r}’)
\n\\end{eqnarray}<\/p>\n

$$\\therefore\\ \\ \\vec{a} = \\vec{a}’ + 2 \\vec{\\omega}\\times\\vec{v}’ + \\vec{\\omega}\\times(\\vec{\\omega}\\times\\vec{r}’)$$<\/p>\n

\u3053\u3053\u3067\uff0c\u4ee5\u4e0b\u3092\u4f7f\u3063\u305f\u3002
\n\\begin{eqnarray}
\n\\dot{\\vec{v}}’ &=& (\\dot{x}’ \\vec{e}’_x + \\dot{y}’ \\vec{e}’_y +\\dot{z}’ \\vec{e}’_z)\\dot{} \\\\
\n&=& (\\ddot{x}’ \\vec{e}’_x + \\ddot{y}’ \\vec{e}’_y +\\ddot{z}’ \\vec{e}’_z) \\\\
\n&&\\quad +(\\dot{x}’ \\dot{\\vec{e}}’_x + \\dot{y}’ \\dot{\\vec{e}}’_y +\\dot{z}’ \\dot{\\vec{e}}’_z) \\\\
\n&=& \\vec{a}’ + \\vec{\\omega}\\times(\\dot{x}’ \\vec{e}’_x + \\dot{y}’ \\vec{e}’_y +\\dot{z}’ \\vec{e}’_z)\\\\
\n&=& \\vec{a}’ + \\vec{\\omega}\\times\\vec{v}’
\n\\end{eqnarray}<\/p>\n

\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u904b\u52d5\u65b9\u7a0b\u5f0f\u3068\u6163\u6027\u529b<\/h3>\n

\u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u3075\u307e\u3048\u3066\uff0c\u56de\u8ee2\u7cfb\u306b\u304a\u3051\u308b\u904b\u52d5\u65b9\u7a0b\u5f0f\u3068\u6163\u6027\u529b\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\u6163\u6027\u7cfb \\(S\\) \u306b\u304a\u3051\u308b\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f
\n$$ m \\vec{a} = \\vec{F}$$ \u5de6\u8fba\u306e\u52a0\u901f\u5ea6 \\(\\vec{a}\\) \u3092 \\(S’\\) \u7cfb\u306e\u901f\u5ea6\u30fb\u52a0\u901f\u5ea6\u3092\u4f7f\u3063\u3066\u8868\u3059\u3068\uff0c
\n$$m \\left( \\vec{a}’ + 2 \\vec{\\omega}\\times\\vec{v}’ + \\vec{\\omega}\\times(\\vec{\\omega}\\times\\vec{r}’)\\right) = \\vec{F}$$ \\(m\\vec{a}’\\) \u4ee5\u5916\u306e\u9805\u3092\u53f3\u8fba\u306b\u79fb\u9805\u3057\u3066
\n$$m\u00a0 \\vec{a}’ = \\vec{F} -2 m\\vec{\\omega}\\times\\vec{v}’ -m \\vec{\\omega}\\times(\\vec{\\omega}\\times\\vec{r}’)$$<\/p>\n

\u56de\u8ee2\u7cfb\u306b\u304a\u3044\u3066\uff08\u65b0\u305f\u306b\uff09\u53f3\u8fba\u306b\u73fe\u308c\u305f\u9805\u3092\uff08\u56de\u8ee2\u7cfb\u3068\u3044\u3046\uff0c\u6163\u6027\u7cfb\u3067\u306f\u306a\u3044\u5ea7\u6a19\u7cfb\u3067\u3042\u3089\u308f\u308c\u308b\u898b\u304b\u3051\u306e\u529b\u3068\u3044\u3046\u610f\u5473\u306e\uff09\u6163\u6027\u529b<\/strong><\/span>\u3068\u8aad\u3093\u3067\u3044\u308b\u3002<\/p>\n

\u4e0a\u8a18\u306e\u6163\u6027\u529b\u306e\u3046\u3061\uff0c\\( -2 m\\vec{\\omega}\\times\\vec{v}’\\) \u306e\u9805\u306f\u56de\u8ee2\u7cfb\u3067\u904b\u52d5\u3057\u3066\u3044\u308b\u3068\u304d\uff0c\u3064\u307e\u308a \\(\\vec{v}’ \\neq \\vec{0}\\) \u306e\u3068\u304d\u306b\u73fe\u308c\u308b\u529b\u3067\u30b3\u30ea\u30aa\u30ea\u529b<\/strong><\/span><\/span>\uff08\u3053\u308a\u304a\u308a\u300e\u304b\u300f\u3067\u306f\u306a\u304f\uff0c\u3053\u308a\u304a\u308a\u300e\u308a\u3087\u304f\u300f\uff01\uff09\u3068\u547c\u3070\u308c\u308b\u3002\u5e38\u306b\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306b\u76f4\u4ea4\u3059\u308b\u6163\u6027\u529b\u3067\u3042\u308a\uff0c\u901f\u5ea6\u306e\u5411\u304d\u3092\u5909\u3048\u308b\u3053\u3068\u304b\u3089\u696d\u754c\u306b\u3088\u3063\u3066\u306f\u8ee2\u5411\u529b<\/strong><\/span>\u3068\u3082\u547c\u3070\u308c\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\\( -m \\vec{\\omega}\\times(\\vec{\\omega}\\times\\vec{r}’)\\) \u306e\u9805\u306f\u56de\u8ee2\u7cfb\u306e\u8ef8\uff08\u4e2d\u5fc3\uff09\u304b\u3089\uff08\u9060\u304f\uff1f\uff09\u96e2\u308c\u305f\u3068\u304d\uff0c\u3064\u307e\u308a \\(\\vec{r}’ \\neq \\vec{0}\\) \u306e\u3068\u304d\u306b\u73fe\u308c\u308b\u529b\u3067\u9060\u5fc3\u529b<\/span><\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n

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