\u6446\u7ebf\u7684\u7b49\u65f6\u6027\u4e0e\u53d8\u5206\u6cd5<\/strong>
\n
\nJQX|Xiao<\/center>
\n\u4e00\uff1a\u6446\u7ebf\u7684\u7b49\u65f6\u6027
\n\u6446\u7ebf\u5728\u5f27\u957f\u65b9\u5411\u6ee1\u8db3\u7b80\u8c10\u632f\u52a8\uff0c\u56e0\u6b64\u4e0e\u5f39\u7c27\u632f\u5b50\u76f8\u540c\uff0c\u9707\u52a8\u5468\u671f\u4e0e\u632f\u5e45\u65e0\u5173\uff0c\u56e0\u6b64\u6ee1\u8db3\u7b49\u65f6\u6027\u3002\u4e0b\u9762\u4ece\u80fd\u91cf\u7684\u89d2\u5ea6\u8bc1\u660e\u6cbf\u6446\u7ebf\u4e0b\u843d\u7684\u7269\u4f53\u6ee1\u8db3\u7b80\u8c10\u632f\u52a8\u3002\u6446\u7ebf\u65b9\u7a0b\u4e3a\uff1a\\(x=R(\\theta-\\sin\\theta),\\qquad y=R(1-\\cos\\theta)\\),\u628a\u6700\u4f4e\u70b9\u4f5c\u4e3a\u52bf\u80fd\u96f6\u70b9\uff1a\u6700\u4f4e\u70b9\u5bf9\u5e94\\(\\theta=\\pi\\)\u3002
\n\u8bb0\u4ece\u6700\u4f4e\u70b9\u5230\u8f68\u9053\u4e0a\u4e00\u70b9P\u7684\u5f27\u957f\u4e3a\\(s\\ge 0\\)\uff0c\u5411\u4e0a\u4e3a\u6b63\uff1bP\u7684\u9ad8\u5ea6\u4e3a\\(h\\ge 0\\)\u3002
\n1) \u5f27\u957f\u5fae\u5206
\n\\( x'(\\theta)=\\frac{dx}{d\\theta}=R(1-\\cos\\theta),\\; y'(\\theta)=\\frac{dy}{d\\theta}=R\\sin\\theta,\\; ds=\\sqrt{(dx)^2+(dy)^2}=\\sqrt{(x’)^2+(y’)^2}\\,d\\theta=2R\\sin\\frac{\\theta}{2}\\,d\\theta. \\)
\n\u56e0\u6b64\u4ece \\( \\theta=\\pi \\) \u5230\u4efb\u610f \\( \\theta \\) \u7684\u5f27\u957f\u4e3a
\n\\( s(\\theta)=\\int_{\\pi}^{\\theta}2R\\sin\\frac{\\varphi}{2}\\,d\\varphi=4R\\cos\\frac{\\theta}{2}. \\)
\n2) \u4ece\u6700\u4f4e\u70b9\u8ba1\u7684\u9ad8\u5ea6
\n\\( h(\\theta)=y(\\theta)-y(\\pi)=R(1-\\cos\\theta)-2R=2R\\cos^2\\frac{\\theta}{2}. \\)
\n3) \\(h\\) \u4e0e \\(s\\) \u7684\u5173\u7cfb
\n\u7531 \\( s=4R\\cos\\frac{\\theta}{2} \\) \u5f97 \\( \\cos\\frac{\\theta}{2}=\\frac{s}{4R} \\)\uff0c\u4ee3\u5165\u4e0a\u5f0f\uff1a
\n\\( h=\\frac{s^{2}}{8R}. \\)
\n4) \u80fd\u91cf\u65b9\u7a0b
\n\u8bbe\u8d28\u70b9\u8d28\u91cf\u4e3a m\uff0c\u5207\u5411\u901f\u5ea6 \\(v=\\dot s\\)\u3002\u80fd\u91cf\u5b88\u6052\uff1a\\(E=\\frac12 m v^2 + m g h =\\frac12 m \\dot s^{\\,2} + m g\\frac{s^2}{8R} =\\frac12 m \\dot s^{\\,2}+\\frac12\\,k’ s^2,\\)\u3002\u5176\u4e2d\\(k’=\\frac{m g}{4R}\\)
\n\u8fd9\u6070\u662f\u4e00\u7ef4\u7b80\u8c10\u632f\u52a8\u7684\u80fd\u91cf\u5f62\u5f0f\uff0c\u89d2\u9891\u7387\\(\\omega=\\sqrt{\\frac{k’}{m}}=\\sqrt{\\frac{g}{4R}}.\\)\u3002\u5229\u7528\u7b80\u8c10\u632f\u52a8\u5468\u671f\u516c\u5f0f\\(T=\\frac{2\\pi}{\\omega}=4\\pi\\sqrt{\\frac{R}{g}},\\qquad t=\\frac{T}{4}=\\pi\\sqrt{\\frac{R}{g}}.\\)
\n\u5468\u671f\u4e0e\u8d77\u70b9\u9ad8\u5ea6\u65e0\u5173\uff0c\u6240\u4ee5\u6446\u7ebf\u6ee1\u8db3\u7b49\u65f6\u6027\u3002
\n\u4e8c\uff1a\u53d8\u5206\u6cd5
\n\u725b\u987f\u53ea\u7528\u4e86\u4e00\u665a\u5c31\u89e3\u51b3\u4e86\u7ea6\u7ff0\u00b7\u4f2f\u52aa\u5229\u5bf9\u4e8e\u6700\u901f\u964d\u7ebf\u7684\u6311\u6218\u7684\u6545\u4e8b\u4e00\u76f4\u4e3a\u4eba\u6d25\u6d25\u4e50\u9053\uff0c\u5f53\u65f6\u5171\u6709\u4e94\u4e2a\u4eba\u7ed9\u51fa\u4e86\u6700\u901f\u964d\u7ebf\u7684\u89e3\u51b3\u529e\u6cd5\u3002\u9664\u4e86\u7ea6\u7ff0\u00b7\u4f2f\u52aa\u5229\u7684\u5de7\u5999\u529e\u6cd5\u5916\uff0c\u4ed6\u7684\u54e5\u54e5\u96c5\u5404\u5e03\u00b7\u4f2f\u52aa\u5229\u66f4\u662f\u5f00\u542f\u4e86\u53d8\u5206\u6cd5\u7684\u5f00\u7aef\uff0c\u51e0\u5341\u5e74\u540e\u6b27\u62c9\u548c\u62c9\u683c\u6717\u65e5\u628a\u8fd9\u4e2a\u65b9\u6cd5\u7cfb\u7edf\u5316\u4e3a\u53d8\u5206\u6cd5\uff0c\u5e76\u63d0\u51fa\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u3002\u4e0b\u9762\u7528E-L\u65b9\u7a0b\u6765\u8bc1\u660e\u6700\u901f\u964d\u7ebf\u4e3a\u6446\u7ebf\u3002
\n1\u3001\u65f6\u95f4\u6cdb\u51fd
\n\u6700\u901f\u964d\u7ebf\u95ee\u9898\u4e3a\u4e00\u4e2a\u5c0f\u7403\u5728\u91cd\u529b\u4f5c\u7528\u4e0b\u4ece\u9ad8\u5904\u6ed1\u5411\u4f4e\u5904\uff0c\u5176\u8fd0\u52a8\u8f68\u8ff9\u66f2\u7ebf\u4e3a \\(y = y(x)\\)\u3002\u6839\u636e\u80fd\u91cf\u5b88\u6052\u5b9a\u5f8b\uff0c\u5c0f\u7403\u5728\u4efb\u610f\u4f4d\u7f6e\u7684\u901f\u5ea6\u4e3a \\(v = \\sqrt{2gy}\\)\u3002\u8bbe\u5fae\u5143\u5f27\u957f\u4e3a \\(dl = \\sqrt{dx^2 + dy^2} = \\sqrt{1 + y’^2}\\,dx\\)\uff0c\u5219\u8d70\u8fc7\u8fd9\u6bb5\u5fae\u5c0f\u5f27\u957f\u6240\u7528\u7684\u65f6\u95f4\u5143\u4e3a \\(dt = \\frac{dl}{v} = \\frac{\\sqrt{1 + y’^2}}{\\sqrt{2gy}}\\, dx\\)\u3002\u56e0\u6b64\uff0c\u5c0f\u7403\u6ed1\u5b8c\u5168\u7a0b\u7684\u603b\u65f6\u95f4\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u4e2a\u79ef\u5206\u6cdb\u51fd \\(T = \\int_{x_1}^{x_2} \\frac{\\sqrt{1 + y’^2}}{\\sqrt{2gy}}\\, dx\\)\uff0c\u6211\u4eec\u8981\u6c42\u89e3\u7684\u5c31\u662f\u80fd\u4f7f T \u6700\u5c0f\u7684\u66f2\u7ebf \\(y(x)\\)\u3002
\n2\u3001\u6c42\u89e3E-L\u65b9\u7a0b
\n\u4e3a\u4e86\u6c42\u51fa\u6781\u503c\u66f2\u7ebf \\(y(x)\\)\uff0c\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u5728\u8fb9\u754c\u70b9\u4e3a\u96f6\u7684\u5fae\u5c0f\u6270\u52a8\u51fd\u6570 \\(\\eta(x)\\)\uff0c\u5e76\u6784\u9020\u4e00\u6761\u6270\u52a8\u540e\u7684\u66f2\u7ebf \\(\\bar{y}(x) = y(x) + \\varepsilon \\eta(x)\\)\u3002\u5176\u4e2d \\(\\varepsilon\\) \u4e3a\u5fae\u5c0f\u5e38\u6570\uff0c\u4e14\u8fb9\u754c\u6761\u4ef6\u6ee1\u8db3 \\(\\eta(x_1) = \\eta(x_2) = 0\\)\uff0c\u4ee5\u4fdd\u8bc1\u6270\u52a8\u4e0d\u6539\u53d8\u66f2\u7ebf\u7684\u7aef\u70b9\u4f4d\u7f6e\u3002
\n\u539f\u65f6\u95f4\u6cdb\u51fd\u53ef\u4ee5\u5199\u4e3a \\(T[y] = \\int_{x_1}^{x_2} F(x, y, y’)\\, dx\\)\uff0c\u5176\u4e2d\u88ab\u79ef\u51fd\u6570\u4e3a \\(F(x, y, y’) = \\frac{\\sqrt{1 + y’^2}}{\\sqrt{2gy}}\\)\u3002\u5c06\u6270\u52a8\u540e\u7684\u66f2\u7ebf \\(\\bar{y} = y + \\varepsilon \\eta\\) \u53ca\u5176\u5bfc\u6570 \\(\\bar{y}’ = y’ + \\varepsilon \\eta’\\) \u4ee3\u5165\uff0c\u6cdb\u51fd\u5c31\u53d8\u6210\u4e86\u5173\u4e8e \\(\\varepsilon\\) \u7684\u51fd\u6570 \\(T(\\varepsilon) = \\int_{x_1}^{x_2} F(x, y + \\varepsilon\\eta, y’ + \\varepsilon\\eta’)\\, dx\\)\u3002
\n\u4e3a\u4e86\u6c42\u5f97\u6cdb\u51fd\u7684\u6781\u503c\uff0c\u6211\u4eec\u9700\u8981\u5176\u5bf9 \\(\\varepsilon\\) \u7684\u4e00\u9636\u53d8\u5206\u4e3a\u96f6\uff0c\u5373 \\(\\frac{dT}{d\\varepsilon}\\Big|_{\\varepsilon=0} = 0\\)\u3002\u9996\u5148\u8ba1\u7b97\u5bfc\u6570\uff1a\\(\\frac{dT}{d\\varepsilon} = \\int_{x_1}^{x_2} \\left( \\frac{\\partial F}{\\partial y}\\eta + \\frac{\\partial F}{\\partial y’}\\eta’ \\right) dx\\)\u3002\u5f53 \\(\\varepsilon = 0\\) \u65f6\uff0c\u6b64\u5f0f\u4f9d\u7136\u6210\u7acb\u3002
\n\u6211\u4eec\u5bf9\u79ef\u5206\u7684\u7b2c\u4e8c\u9879\u4f7f\u7528\u5206\u90e8\u79ef\u5206\u6cd5\uff08\\(\\int u\\,dv=uv-\\int v\\,du\\)\uff09\u3002\u4ee4 \\(u=\\frac{\\partial F}{\\partial y’}\\) \u4e14 \\(dv=\\eta’\\,dx\\)\uff0c\u5219\u53ef\u5f97\u5230 \\(v=\\eta\\) \u548c \\(du=\\frac{d}{dx}\\!\\left(\\frac{\\partial F}{\\partial y’}\\right)dx\\)\u3002\u4e8e\u662f\uff0c\u79ef\u5206\u9879\u53d8\u4e3a \\(\\int_{x_1}^{x_2}\\frac{\\partial F}{\\partial y’}\\,\\eta’\\,dx =\\left[\\frac{\\partial F}{\\partial y’}\\,\\eta\\right]_{x_1}^{x_2} -\\int_{x_1}^{x_2}\\eta\\,\\frac{d}{dx}\\!\\left(\\frac{\\partial F}{\\partial y’}\\right)dx\\)\u3002
\n\u7531\u4e8e\u8fb9\u754c\u5904\u7684\u6270\u52a8\u4e3a\u96f6\uff0c\u5373 \\(\\eta(x_1)=\\eta(x_2)=0\\)\uff0c\u6240\u4ee5 \\(\\left[\\frac{\\partial F}{\\partial y’}\\,\\eta\\right]_{x_1}^{x_2}\\) \u8fd9\u4e00\u9879\u4e3a\u96f6\u3002\u6700\u7ec8\u6211\u4eec\u5f97\u5230 \\(\\frac{dT}{d\\varepsilon}\\Big|_{\\varepsilon=0} = \\int_{x_1}^{x_2}\\!\\left( \\frac{\\partial F}{\\partial y} -\\frac{d}{dx}\\!\\left(\\frac{\\partial F}{\\partial y’}\\right) \\right)\\eta\\,dx=0\\)\u3002
\n\u6839\u636e\u53d8\u5206\u6cd5\u57fa\u672c\u5f15\u7406\uff0c\u56e0\u4e3a \\(\\eta(x)\\) \u662f\u4efb\u610f\u7684\u5fae\u5c0f\u6270\u52a8\uff0c\u8981\u4f7f\u4e0a\u5f0f\u6052\u6210\u7acb\uff0c\u5fc5\u987b\u5176\u62ec\u53f7\u5185\u7684\u90e8\u5206\u4e3a\u96f6\u3002\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u8457\u540d\u7684\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\uff1a\\(\\frac{\\partial F}{\\partial y} -\\frac{d}{dx}\\!\\left(\\frac{\\partial F}{\\partial y’}\\right)=0\\)\u3002
\n3\u3001Beltrami\u6052\u7b49\u5f0f\u53ca\u5176\u5e94\u7528
\n\u5728\u67d0\u4e9b\u7279\u6b8a\u60c5\u51b5\u4e0b\uff0c\u6b27\u62c9-\u62c9\u683c\u6717\u65af\u65b9\u7a0b\u53ef\u4ee5\u88ab\u7b80\u5316\u3002\u4e00\u4e2a\u91cd\u8981\u7684\u60c5\u5f62\u662f\u5f53\u88ab\u79ef\u51fd\u6570 \\(F(x, y, y’)\\) \u4e0d\u663e\u5f0f\u5730\u4f9d\u8d56\u4e8e \\(x\\) \u65f6\uff08\u5373 \\(\\frac{\\partial F}{\\partial x} = 0\\)\uff09\uff0c\u5b58\u5728\u4e00\u4e2a\u88ab\u79f0\u4e3a**\u8d1d\u5c14\u7279\u62c9\u7c73\u6052\u7b49\u5f0f (Beltrami Identity)** \u7684\u4e00\u9636\u79ef\u5206\uff1a\\(F – y’ \\frac{\\partial F}{\\partial y’} = C\\)\uff0c\u5176\u4e2d C \u662f\u4e00\u4e2a\u5e38\u6570\u3002\u8fd9\u4e2a\u6052\u7b49\u5f0f\u53ef\u4ee5\u5927\u5927\u7b80\u5316\u6c42\u89e3\u8fc7\u7a0b\u3002
\n\u73b0\u5728\uff0c\u6211\u4eec\u628a\u8fd9\u4e2a\u5f3a\u5927\u7684\u5de5\u5177\u5e94\u7528\u5230\u6700\u901f\u964d\u7ebf\u95ee\u9898\u4e0a\u3002\u56de\u987e\u6211\u4eec\u7684\u65f6\u95f4\u6cdb\u51fd\u88ab\u79ef\u51fd\u6570 \\(L = \\frac{\\sqrt{1 + y’^2}}{\\sqrt{2gy}}\\)\uff0c\u5b83\u663e\u7136\u4e0d\u663e\u5f0f\u5730\u542b\u6709\u53d8\u91cf \\(x\\)\uff0c\u56e0\u6b64\u5b8c\u5168\u6ee1\u8db3\u4f7f\u7528\u8d1d\u5c14\u7279\u62c9\u7c73\u6052\u7b49\u5f0f\u7684\u6761\u4ef6\u3002
\n\u6211\u4eec\u9996\u5148\u8ba1\u7b97 \\(L\\) \u5bf9 \\(y’\\) \u7684\u504f\u5bfc\u6570\uff1a\\(\\frac{\\partial L}{\\partial y’} =\\frac{1}{\\sqrt{2g}}\\frac{y’}{\\sqrt{y}\\sqrt{1+y’^2}}\\)\u3002\u7136\u540e\u5c06 \\(L\\) \u548c \\(\\frac{\\partial L}{\\partial y’}\\) \u4ee3\u5165\u6052\u7b49\u5f0f \\(L-y’\\,\\frac{\\partial L}{\\partial y’}=C\\)\u3002\u7ecf\u8fc7\u5316\u7b80\uff0c\u6211\u4eec\u5f97\u5230\u4e00\u4e2a\u975e\u5e38\u7b80\u6d01\u7684\u5173\u7cfb\u5f0f\uff1a\\(\\frac{1}{\\sqrt{2g}}\\frac{1}{\\sqrt{y}\\sqrt{1+y’^2}}=C\\)\u3002
\n4\u3001\u6c42\u89e3\u65b9\u7a0b\uff0c\u8bc1\u660e\u66f2\u7ebf\u4e3a\u6446\u7ebf
\n\u4e3a\u4e86\u6c42\u89e3\u4e0a\u9762\u8fd9\u4e2a\u5173\u4e8e \\(y\\) \u548c \\(y’\\) \u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u6211\u4eec\u9996\u5148\u5c06\u6240\u6709\u5e38\u6570\u5408\u5e76\u4e3a\u4e00\u4e2a\u65b0\u7684\u5e38\u6570 \\(k\\)\uff0c\u5f97\u5230 \\(\\frac{1}{\\sqrt{y}\\sqrt{1+y’^2}}=k\\)\u3002\u6574\u7406\u540e\u53ef\u89e3\u51fa \\(y’\\) \u7684\u8868\u8fbe\u5f0f\uff1a\\(y’^2=\\frac{1-k^2y}{k^2y}\\)\u3002
\n\u4e3a\u4e86\u5bf9\u8be5\u65b9\u7a0b\u79ef\u5206\uff0c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a\u6280\u5de7\uff0c\u6c42\u89e3 \\(x\\) \u5173\u4e8e \\(y\\) \u7684\u5bfc\u6570 \\(\\frac{dx}{dy}=\\frac{1}{y’}=\\sqrt{\\frac{k^2 y}{1-k^2 y}}\\)\u3002\u8fd9\u4e2a\u5f62\u5f0f\u7684\u79ef\u5206\u53ef\u4ee5\u901a\u8fc7\u53c2\u6570\u4ee3\u6362\u6765\u89e3\u51b3\u3002\u6211\u4eec\u5f15\u5165\u53c2\u6570 \\(\\theta\\)\uff0c\u5e76\u4ee4 \\(y=\\frac{1}{k^2}\\sin^2\\!\\frac{\\theta}{2}\\)\uff0c\u5b83\u7b49\u4ef7\u4e8e \\(y=\\frac{1-\\cos\\theta}{2k^2}\\)\u3002
\n\u8fdb\u884c\u4ee3\u6362\u540e\uff0c\u8868\u8fbe\u5f0f\u88ab\u7b80\u5316\u4e3a\uff1a\\(\\frac{dx}{dy}=\\sqrt{\\frac{(1-\\cos\\theta)\/2}{(1+\\cos\\theta)\/2}} =\\tan\\frac{\\theta}{2}\\)\u3002\u540c\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6c42\u51fa \\(dy=\\frac{1}{2k^2}\\sin\\theta\\,d\\theta\\)\uff0c\u56e0\u6b64 \\(dx=\\tan\\frac{\\theta}{2}\\,dy = \\frac{1}{k^2}\\sin^2\\!\\frac{\\theta}{2}\\,d\\theta\\)\u3002\u5bf9 \\(dx\\) \u79ef\u5206\u53ef\u5f97 \\(x=\\frac{1}{2k^2}\\int (1-\\cos\\theta)\\,d\\theta =\\frac{1}{2k^2}\\bigl(\\theta-\\sin\\theta\\bigr)+C_x\\)\uff0c\u5176\u4e2d \\(C_x\\) \u662f\u79ef\u5206\u5e38\u6570\u3002
\n\u6700\u540e\uff0c\u6211\u4eec\u6574\u7406\u4e00\u4e0b\u7ed3\u679c\u3002\u4ee4\u5e38\u6570\u534a\u5f84 \\(R=\\frac{1}{2k^2}\\)\uff0c\u5e76\u5c06\u79ef\u5206\u5e38\u6570\u5408\u5e76\u4e3a\u8d77\u70b9\u5750\u6807 \\((x_0, y_0)\\)\uff0c\u6211\u4eec\u4fbf\u5f97\u5230\u4e86\u8be5\u66f2\u7ebf\u7684\u6700\u7ec8\u53c2\u6570\u65b9\u7a0b\uff1a
\n\\(\\begin{aligned} x(\\theta) &= R\\bigl(\\theta-\\sin\\theta\\bigr) \\\\ y(\\theta) &= R\\bigl(1-\\cos\\theta\\bigr) \\end{aligned}\\)
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