![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-640x308.png\")
<\/p>\n
<\/p>\n
\u30ab\u30c6\u30ca\u30ea\u30fc\u65b9\u7a0b\u5f0f<\/h3>\n
\u4e00\u69d8\u91cd\u529b\u5834\u4e2d\u306b\u4e00\u69d8\u306a\u8cea\u91cf\u7dda\u5bc6\u5ea6\u306e\u30ed\u30fc\u30d7\uff08\u30ef\u30a4\u30e4\u30fc\uff0c\u96fb\u7dda\u7b49\u306a\u3093\u3067\u3082\u3088\u3044\uff09\u3092\u4e21\u7aef\u3092\u56fa\u5b9a\u3057\u3066\u5782\u3089\u3057\u305f\u3068\u304d\u306b\u3067\u304d\u308b\u66f2\u7dda\u304c\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda<\/strong><\/span>\uff08\u61f8\u5782\u66f2\u7dda<\/strong><\/span>\uff0c\u61f8\u5782\u7dda<\/strong><\/span>\u3068\u3082\uff09\u3067\u3042\u308a\uff0c\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda<\/strong><\/span>\u304c\u6e80\u305f\u3059\u65b9\u7a0b\u5f0f<\/p>\n $$\\frac{d^2 y}{dx^2} = a \\sqrt{1 + \\left( \\frac{dy}{dx} \\right)^2}$$<\/p>\n \u306f\u516c\u5f0f\u306b\u306f\u540d\u524d\u304c\u3064\u3044\u3066\u3044\u306a\u3044\u3088\u3046\u3060\u304c\uff0c\u3053\u3053\u3067\u306f\u30ab\u30c6\u30ca\u30ea\u30fc\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u547c\u3076\u3053\u3068\u306b\u3057\u305f\u306e\u3067\u3042\u3063\u305f\u3002<\/a><\/p>\n \u30ed\u30fc\u30d7\u7d20\u7247\u306b\u306f\u305f\u3089\u304f\u529b\u306e\u3064\u308a\u3042\u3044\u304b\u3089\u306e\u30ab\u30c6\u30ca\u30ea\u30fc\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa\u306b\u3064\u3044\u3066\u306f\uff0c\u5225\u30da\u30fc\u30b8\u306b\u8a18\u8f09\u6e08\u307f<\/a>\u3002\u3053\u3053\u3067\u306f\uff0c\u5225\u89e3\u3068\u3057\u3066\u5909\u5206\u539f\u7406\u306b\u3088\u308b\u5c0e\u51fa\u3092\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n Wikipedia \u82f1\u8a9e\u7248<\/a>\u306a\u3069\u3067\u306f\uff0c\u30ed\u30fc\u30d7\u306e\u5168\u9577 $$L \\equiv \\int_A^B ds$$ \u304c\u4e00\u5b9a\u3068\u3044\u3046\u62d8\u675f\u6761\u4ef6\u306e\u3082\u3068\u3067\uff0c\u5168\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc $$U \\equiv \\int_A^B \\rho g y\\, ds$$ \u3092\u6700\u5c0f\u306b\u3059\u308b\u3053\u3068\u304b\u3089\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda\u3092\u5c0e\u304f\u3068\u3057\u3066\u3044\u308b\u304c\uff0c\u5b9f\u969b\u306b\u306f\u62d8\u675f\u6761\u4ef6\u306a\u3057\u3067 $\\delta U = 0$ \u3068\u3044\u3046\u5909\u5206\u539f\u7406\u3060\u3051\u304b\u3089\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n \u307e\u305a\uff0c\u9577\u3055 $ds = \\sqrt{dx^2 + dy^2}$ \u306e\u30ed\u30fc\u30d7\u7d20\u7247\u306e\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc $dU$ \u306f\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u3092 $g$\uff0c\u30ed\u30fc\u30d7\u306e\u8cea\u91cf\u7dda\u5bc6\u5ea6 $\\rho$ \u3092\u4e00\u5b9a\uff0c\u6c34\u5e73\u65b9\u5411\u3092 $x$\uff0c\u5782\u76f4\u65b9\u5411\u3092 $y$\uff0c\u30ed\u30fc\u30d7\u4e0a\u306e\u4f4d\u7f6e\u3092 $(x, y)$ \u3068\u3057\u3066<\/p>\n $$dU = \\rho g y \\, ds = \\rho g y \\sqrt{1 + \\left( \\frac{dy}{dx}\\right)^2}\\, dx \\equiv \\rho g y \\sqrt{1 + \\left( y’\\right)^2} \\, dx$$<\/p>\n \u30ed\u30fc\u30d7\u306e\u4e21\u7aef $A, B$ \u3092\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\u5168\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc $U$ \u306f<\/p>\n $$U =\\int_A^B dU = \\rho g \\int_A^B y \\sqrt{1 + \\left( y’\\right)^2} \\, dx$$<\/p>\n $\\delta U = 0$ \u304b\u3089\uff0c$f(y, y’) \\equiv y \\sqrt{1 + \\left( y’\\right)^2} $ \u306b\u5bfe\u3059\u308b\u30aa\u30a4\u30e9\u30fc\u30fb\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f\u306f<\/p>\n $$\\frac{\\partial f}{\\partial y} – \\frac{d}{dx} \\left( \\frac{\\partial f}{\\partial y’}\\right) = 0$$<\/p>\n \u3053\u306e\u307e\u307e\u8a08\u7b97\u3092\u7d9a\u3051\u3066\u3044\u304f\u3068 $y y^{”} = 1 + \\left( y’\\right)^2$ \u3068\u3044\u3046\u65b9\u7a0b\u5f0f\u304c\u51fa\u3066\u304f\u308b\u3002\u3053\u308c\u3067\u3082\u89e3\u304c $y = \\frac{1}{a} \\cosh(ax)$ \u306e\u5f62\u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u308b\u304c\uff0c\u30aa\u30a4\u30e9\u30fc\u30fb\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f\u306b $y’$ \u3092\u304b\u3051\u3066\u30d9\u30eb\u30c8\u30e9\u30df\u306e\u516c\u5f0f<\/strong><\/span>\u306e\u5f62\u306b\u3057\u3066\u307f\u308b\u3002<\/p>\n \\begin{eqnarray} \u3064\u307e\u308a\uff0c<\/p>\n \\begin{eqnarray} \u3053\u308c\u304c\u30d9\u30eb\u30c8\u30e9\u30df\u306e\u516c\u5f0f<\/span><\/strong>\u3002\u5b9a\u6570\u3092 $\\frac{1}{a}$ \u3068\u3057\u305f\u306e\u306f\uff0c\u5358\u306b\u6211\u3005\u304c\u30ab\u30c6\u30ca\u30ea\u30fc\u65b9\u7a0b\u5f0f\u3068\u547c\u3093\u3067\u3044\u308b\u3082\u306e\u306e\u5f62\u306b\u3042\u308f\u305b\u308b\u305f\u3081\u3002\u3042\u3068\u306f\uff0c$f(y,y’) = y \\sqrt{1 + \\left( y’\\right)^2}$ \u3092\u4ee3\u5165\u3057\u3066\u8a08\u7b97\u3057\u3066\u3044\u304f\u3068\uff0c<\/p>\n \\begin{eqnarray} \u6700\u5f8c\u306e\u5f0f\u306e\u4e21\u8fba\u3092 $x$ \u3067\u3082\u3046\u4e00\u5ea6\u5fae\u5206\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u3068\u306a\u308a\uff0c\u7121\u4e8b\u30ab\u30c6\u30ca\u30ea\u30fc\u65b9\u7a0b\u5f0f<\/strong><\/span>\u304c\u5c0e\u304b\u308c\u305f\u3002\u3042\u3068\u306f\uff0c\u5225\u30da\u30fc\u30b8\u306e\u89e3\u6cd5<\/a>\u306b\u5f93\u3048\u3070\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda\u304c\u5c0e\u304b\u308c\u308b\u3002<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" \u5168\u91cd\u529b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc $\\displaystyle U = \\int_A^B \\rho g y\\, ds$ \u3092\u6700\u5c0f\u306b\u3059\u308b\u66f2\u7dda\u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\u30ab\u30c6\u30ca\u30ea\u30fc\u65b9\u7a0b\u5f0f\u3092\u5c0e\u304f\u3002\u4ee5\u4e0b\u3082\u53c2\u7167\uff1a<\/p>\u5909\u5206\u539f\u7406\u306b\u3088\u308b\u5c0e\u51fa<\/h3>\n
\ny’ \\left\\{ \\frac{\\partial f}{\\partial y} – \\frac{d}{dx} \\left( \\frac{\\partial f}{\\partial y’}\\right) \\right\\} &=&
\ny’ \\frac{\\partial f}{\\partial y} – \\left\\{\\frac{d}{dx}\\left(y’ \\frac{\\partial f}{\\partial y’} \\right) – \\frac{d y’}{dx} \\frac{\\partial f}{\\partial y’}\\right\\} \\\\
\n&=& \\left(\\frac{dy}{dx} \\frac{\\partial}{\\partial y} + \\frac{d y’}{dx} \\frac{\\partial}{\\partial y’} \\right) f(y,y’) – \\frac{d}{dx}\\left(y’ \\frac{\\partial f}{\\partial y’} \\right) \\\\
\n&=& \\frac{d}{dx} f(y,y’) – \\frac{d}{dx}\\left(y’ \\frac{\\partial f}{\\partial y’} \\right) \\\\
\n&=& \\frac{d}{dx} \\left( f – y’ \\frac{\\partial f}{\\partial y’}\\right) \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n
\n\\frac{d}{dx} \\left( f – y’ \\frac{\\partial f}{\\partial y’}\\right) &=& 0 \\\\
\n\\therefore\\ \\ f – y’ \\frac{\\partial f}{\\partial y’} &=& \\mbox{const.} \\equiv \\frac{1}{a}
\n\\end{eqnarray}<\/p>\n
\ny \\sqrt{1 + \\left(y’\\right)^2} – y y’ \\frac{y’}{\\sqrt{1 + \\left(y’\\right)^2}} &=& \\frac{1}{a} \\\\
\n\\frac{y}{\\sqrt{1 + \\left(y’\\right)^2}} &=& \\frac{1}{a} \\\\
\n\\therefore\\ \\ a^2 y^2 &=& 1 + \\left(y’\\right)^2
\n\\end{eqnarray}<\/p>\n
\n\\frac{d}{dx} \\left( 1 + \\left(y’\\right)^2\\right) &=& \\frac{d}{dx}\\left( a^2 y^2 \\right) \\\\
\n2 y’ y^{”} &=& 2 a^2 y y’ \\\\
\n\\therefore \\ \\ y^{”}\u00a0 &=& a^2 y \\\\
\n&=& a \\sqrt{1 + \\left(y’\\right)^2} \\\\ \\ \\\\
\n\\therefore\\ \\ \\frac{d^2 y}{dx^2} &=& a \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}
\n\\end{eqnarray}<\/p>\n