{"id":28768,"date":"2024-09-19T17:27:48","date_gmt":"2024-09-19T17:27:48","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28768"},"modified":"2024-09-19T17:31:40","modified_gmt":"2024-09-19T17:31:40","slug":"%e5%8f%98%e5%88%86%e6%b3%95%e4%b8%ad%e7%9a%84%e6%9c%80%e9%80%9f%e9%99%8d%e7%ba%bf%e4%b8%8e%e6%ac%a7%e6%8b%89-%e6%8b%89%e6%a0%bc%e6%9c%97%e6%97%a5%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e5%8f%98%e5%88%86%e6%b3%95%e4%b8%ad%e7%9a%84%e6%9c%80%e9%80%9f%e9%99%8d%e7%ba%bf%e4%b8%8e%e6%ac%a7%e6%8b%89-%e6%8b%89%e6%a0%bc%e6%9c%97%e6%97%a5%e6%96%b9%e7%a8%8b\/","title":{"rendered":"\u53d8\u5206\u6cd5\u4e2d\u7684\u6700\u901f\u964d\u7ebf\u4e0e\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b"},"content":{"rendered":"<h1 style=\"text-align:center;\">\u7ecf\u5178\u529b\u5b66\u4e2d\u7684\u53d8\u5206\u6cd5\u4e0e\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b<\/h1>\n<p style=\"text-align:center;\"><em><strong>\u6458\u8981\uff1a<\/strong><\/br>\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u53d8\u5206\u6cd5\u6280\u672f\u63a8\u5bfc\u51fa\u5206\u6790\u529b\u5b66\u4e2d\u7684\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\uff0c\u5e76\u8be6\u7ec6\u5c55\u793a\u5176\u5728\u89e3\u51b3\u6700\u901f\u964d\u7ebf\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002<\/em><\/p>\n<hr>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5b8c\u6210\u672c\u8282\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol  style=\"text-align:left;\">\n<li><strong>\u7406\u89e3<\/strong>\u54c8\u5bc6\u987f\u7684\u6700\u5c0f\u4f5c\u7528\u539f\u7406<\/li>\n<li><strong>\u63a8\u5bfc<\/strong>\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b<\/li>\n<li><strong>\u89e3\u51b3<\/strong>\u4f7f\u7528\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u89e3\u51b3\u6700\u901f\u964d\u7ebf\u95ee\u9898\u3002<\/li>\n<\/ol>\n<hr>\n<p style=\"text-align:center;\">\n<strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>:<\/strong><br \/>\n<a href=\"#1\">\u7ecf\u5178\u529b\u5b66\u4e2d\u4e3a\u4f55\u4f7f\u7528\u53d8\u5206\u6cd5<\/a><br \/>\n<a href=\"#2\">\u53d8\u5206\u95ee\u9898\u7684\u8868\u8ff0<\/a><br \/>\n<a href=\"#3\">\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b<\/a><br \/>\n<a href=\"#4\">\u6700\u901f\u964d\u7ebf\u95ee\u9898<\/a><br \/>\n<a href=\"#5\">\u5305\u542b Wolfram \u7b97\u6cd5\u7684 GitHub \u4ed3\u5e93<\/a>\n<\/p>\n<hr>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/JyumifjGzM0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><\/p>\n<hr>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u7ecf\u5178\u529b\u5b66\u4e2d\u4e3a\u4f55\u4f7f\u7528\u53d8\u5206\u6cd5<\/h2>\n<p style=\"text-align:justify;\">\u725b\u987f\u7269\u7406\u5b66\u4e2d\u5b58\u5728\u8bb8\u591a\u95ee\u9898\uff0c\u53ef\u4ee5\u66f4\u6709\u6548\u5730\u4f7f\u7528\u53d8\u5206\u6cd5\u6765\u5904\u7406\u3002\u8fd9\u79cd\u65b9\u6cd5\u5728\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u548c\u54c8\u5bc6\u987f\u7684\u6700\u5c0f\u4f5c\u7528\u539f\u7406\u4e2d\u8d77\u5230\u5173\u952e\u4f5c\u7528\u3002\u672c\u8d28\u4e0a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u662f\u5bfb\u627e\u6700\u5927\u5316\u6216\u6700\u5c0f\u5316\u67d0\u4e2a\u91cf\u7684\u8def\u5f84\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u5bfb\u627e\u4e24\u70b9\u4e4b\u95f4\u7684\u8def\u5f84\uff0c\u4ee5\u6700\u5c0f\u5316\u884c\u9a76\u8ddd\u79bb\u6216\u65f6\u95f4\u3002\u8be5\u65b9\u6cd5\u7684\u4e00\u4e2a\u4f8b\u5b50\u662f\u8d39\u9a6c\u539f\u7406\uff0c\u8be5\u539f\u7406\u6307\u51fa\u5149\u603b\u662f\u6cbf\u7740\u6700\u5c0f\u5316\u884c\u8fdb\u65f6\u95f4\u7684\u8def\u5f84\u4f20\u64ad\uff0c\u8fd9\u53cd\u8fc7\u6765\u53c8\u5bfc\u81f4\u4e86<a href=\"http:\/\/toposuranos.com\/material\/zh\/%e5%85%89%e7%9a%84%e6%8a%98%e5%b0%84%e5%92%8c%e6%96%af%e6%b6%85%e5%b0%94%e5%ae%9a%e5%be%8b\/\" target=\"_blank\" rel=\"noopener\">\u65af\u6d85\u5c14\u5b9a\u5f8b\uff08\u5149\u7684\u6298\u5c04\u5b9a\u5f8b\uff09<\/a>\u3002<\/p>\n<p style=\"text-align:justify;\">\u53d8\u5206\u6cd5\u5728\u7ecf\u5178\u529b\u5b66\u4e2d\u5177\u6709\u591a\u79cd\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5b83\u53ef\u4ee5\u4e3a\u5177\u6709\u5bf9\u79f0\u6027\u7684\u7cfb\u7edf\u63d0\u4f9b\u7cbe\u786e\u7684\u89e3\u6790\u89e3\uff0c\u5e76\u901a\u8fc7\u53d8\u5206\u6270\u52a8\u7406\u8bba\u4e3a\u66f4\u590d\u6742\u7684\u7cfb\u7edf\u63d0\u4f9b\u8fd1\u4f3c\u89e3\u3002\u6b64\u5916\uff0c\u5728\u96be\u4ee5\u7528\u5fae\u5206\u65b9\u7a0b\u8868\u8fbe\u529b\u7684\u60c5\u51b5\u4e0b\uff0c\u6700\u5c0f\u4f5c\u7528\u539f\u7406\u63d0\u4f9b\u4e86\u4e00\u79cd\u66f4\u6709\u6548\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u7ecf\u5178\u529b\u5b66\u95ee\u9898\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u53d8\u5206\u6cd5\u662f\u4e00\u79cd\u91cd\u8981\u7684\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u725b\u987f\u5b9a\u5f8b\u7684\u66ff\u4ee3\u8868\u8ff0\u3001\u7269\u7406\u5b9a\u5f8b\u7684\u7edf\u4e00\u3001\u66f4\u9ad8\u6548\u7684\u95ee\u9898\u89e3\u51b3\u65b9\u6cd5\uff0c\u4ee5\u53ca\u66f4\u51c6\u786e\u7684\u5b9e\u9a8c\u7ed3\u679c\u9884\u6d4b\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u53d8\u5206\u95ee\u9898\u7684\u8868\u8ff0<\/h2>\n<p style=\"text-align:justify;\">\u53d8\u5206\u6cd5\u7684\u6838\u5fc3\u662f\u627e\u5230\u4f7f\u6cdb\u51fd\u503c\u6781\u503c\u5316\u7684\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span>\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">J(x,y(x))=\\displaystyle \\int_{x_1}^{x_2} f\\left(x,y(x),\\frac{dy(x)}{dx}\\right)dx,<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u627e\u5230\u5176\u6700\u5927\u503c\u6216\u6700\u5c0f\u503c\u3002\u5728\u8fd9\u4e2a\u65b9\u7a0b\u4e2d\uff0c\u6cdb\u51fd <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u4f9d\u8d56\u4e8e\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u53ca\u5176\u5bfc\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dy(x)\/dx,<\/span><\/span> \u540c\u65f6\u79ef\u5206\u4e0a\u4e0b\u9650\u4fdd\u6301\u56fa\u5b9a\u3002\u4e3a\u4e86\u6781\u503c\u5316\u79ef\u5206\uff0c\u5bf9\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u65bd\u52a0\u53d8\u5316\uff0c\u4ee5\u627e\u5230\u4f7f\u6cdb\u51fd\u8fbe\u5230\u6781\u503c\u7684\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u5982\u679c\u4f7f\u79ef\u5206\u8fbe\u5230\u6700\u5c0f\u503c\uff0c\u90a3\u4e48\u5176\u90bb\u57df\u5185\u7684\u4efb\u4f55\u51fd\u6570\uff0c\u65e0\u8bba\u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u6709\u591a\u63a5\u8fd1\uff0c\u90fd\u4f1a\u589e\u52a0\u6cdb\u51fd\u7684\u503c\u3002<\/p>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u5efa\u7acb\u201c\u90bb\u57df\u51fd\u6570\u201d\u7684\u6982\u5ff5\uff0c\u6211\u4eec\u53ef\u4ee5\u7ed9\u6240\u6709\u53ef\u80fd\u7684\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u5206\u914d\u4e00\u4e2a\u53c2\u6570\u8868\u793a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=(\\alpha,x)<\/span><\/span>\uff0c\u8fd9\u6837\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=0<\/span><\/span>\uff0c\u90a3\u4e48 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(0,x)=y(x)<\/span><\/span> \u5c31\u662f\u4f7f <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u6781\u503c\u5316\u7684\u51fd\u6570\u3002\u8fd9\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(\\alpha, x) = y(x) + \\alpha \\eta(x),<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x)<\/span><\/span> \u662f\u4e00\u4e2a\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_2<\/span><\/span> \u5904\u4e3a\u96f6\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}^1<\/span><\/span> \u7c7b\u51fd\u6570\uff0c\u56e0\u6b64\u5305\u62ec\u8be5\u53d8\u5316\u7684\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(\\alpha,x)<\/span><\/span> \u5728\u79ef\u5206\u8def\u5f84\u7684\u8d77\u70b9\u548c\u7ec8\u70b9\u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u76f8\u540c\u3002<\/p>\n<p style=\"text-align:justify;\">\u901a\u8fc7\u5c06\u5305\u62ec\u53d8\u5316 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x)<\/span><\/span> \u7684\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(\\alpha,x)<\/span><\/span> \u66ff\u6362\u8fdb\u6cdb\u51fd <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u5b9a\u4e49\u7684\u79ef\u5206\u4e2d\uff0c\u800c\u4e0d\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span>\uff0c\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u4f9d\u8d56\u4e8e\u53c2\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> \u7684\u6cdb\u51fd\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">J(x,y(\\alpha, x)) = \\displaystyle \\int_{x_1}^{x_2} f\\left(x,y(\\alpha,x), \\dfrac{d}{dx}y(\\alpha,x)\\right)dx<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u5b58\u5728\u5c40\u90e8\u6781\u503c\uff0c\u5fc5\u987b\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\dfrac{\\partial J(x,y(\\alpha,x))}{\\partial \\alpha}\\right|_{\\alpha=0} = 0<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u5bf9\u4e8e\u4efb\u4f55\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x).<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b<\/h2>\n<p style=\"text-align:justify;\">\u901a\u8fc7\u5206\u6790\u5bfc\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\partial J(x,y(\\alpha,x))\/\\partial \\alpha<\/span><\/span>\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n{}\\dfrac{\\partial J(x,y(\\alpha,x))}{\\partial \\alpha}\n\n&amp;=&amp;\\dfrac{\\partial}{\\partial \\alpha} \\displaystyle \\int_{x_1}^{x_2} f\\left(x,y(\\alpha,x),\\dfrac{dy(\\alpha, x)}{dx}\\right)dx \\\\ \\\\\n\n&amp;=&amp;\\displaystyle \\int_{x_1}^{x_2} \\left(\\dfrac{\\partial f}{\\partial x}\\dfrac{\\partial x}{\\partial  \\alpha} + \\dfrac{\\partial f}{\\partial y(\\alpha, x)}\\dfrac{\\partial y(\\alpha, x)}{\\partial  \\alpha}  + \\dfrac{\\partial f }{ \\partial \\frac{dy(\\alpha,x)}{dx}} \\dfrac{\\partial \\frac{dy(\\alpha,x)}{dx}}{\\partial \\alpha}\\right)dx \\\\\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">\u4ece\u8fd9\u4e00\u70b9\u51fa\u53d1\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u51e0\u70b9\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\\dfrac{\\partial x}{\\partial \\alpha} &amp;=&amp; 0 \\\\ \\\\\n\n\\dfrac{\\partial y(\\alpha,x)}{\\partial \\alpha} &amp;=&amp; \\dfrac{\\partial}{\\partial \\alpha} \\left(y(x) + \\alpha \\eta(x) \\right) = \\eta(x) \\\\ \\\\\n\n\\dfrac{\\partial}{\\partial \\alpha}\\left( \\dfrac{dy(\\alpha, x)}{dx} \\right)&amp;=&amp; \\dfrac{\\partial}{\\partial \\alpha} \\left(\\dfrac{dy(x)}{dx} + \\alpha\\dfrac{d\\eta(x)}{dx} \\right) = \\dfrac{d\\eta}{dx}\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">\u56e0\u6b64\uff0c\u8868\u8fbe\u5f0f\u53ef\u4ee5\u7b80\u5316\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\dfrac{\\partial J(x,y(\\alpha,x))}{\\partial \\alpha} &amp;=&amp; \\displaystyle \\int_{x_1}^{x_2} \\left(\\dfrac{\\partial f}{\\partial y(\\alpha,x)}\\eta(x) + \\dfrac{\\partial f}{\\partial  \\frac{dy(\\alpha,x)}{dx}} \\dfrac{d\\eta(x)}{dx} \\right)dx \\\\ \\\\\n\n&amp;=&amp;\\displaystyle \\int_{x_1}^{x_2} \\dfrac{\\partial f}{\\partial y(\\alpha,x)}\\eta(x) dx +  \\int_{x_1}^{x_2} \\dfrac{\\partial f}{\\partial  \\frac{dy(\\alpha,x)}{dx}} \\dfrac{d\\eta(x)}{dx} dx\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">\u7136\u540e\uff0c\u5982\u679c\u6211\u4eec\u89c2\u5bdf\u7b2c\u4e8c\u4e2a\u79ef\u5206\uff0c\u5c06\u4f1a\u53d1\u73b0\u53ef\u4ee5\u901a\u8fc7\u5206\u90e8\u79ef\u5206\u6765\u7b80\u5316\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\\displaystyle \\int_{x_1}^{x_2} \\dfrac{\\partial f}{\\partial  \\frac{dy(\\alpha,x)}{dx}} \\dfrac{d\\eta}{dx} dx\n\n&amp;=&amp; \\left. \\dfrac{\\partial f}{\\partial \\frac{dy(\\alpha,x)}{dx}} \\eta(x)\\right|_{x_1}^{x_2} - \\displaystyle \\int_{x_1}^{x_2}\\eta(x) \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial \\frac{dy(\\alpha, x)}{dx}} \\right) dx\\\\ \\\\\n\n&amp;=&amp; - \\displaystyle \\int_{x_1}^{x_2}\\eta(x) \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial \\frac{dy(\\alpha, x)}{dx}} \\right)dx\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n{} \\dfrac{\\partial J(x,y(\\alpha,x))}{\\partial \\alpha}\n\n&amp;=&amp; \\displaystyle \\int_{x_1}^{x_2} \\left[ \\eta(x) \\dfrac{\\partial f}{\\partial y(\\alpha, x)}  - \\eta(x) \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial \\frac{dy(\\alpha,x)}{dx}} \\right) \\right]dx \\\\ \\\\\n\n&amp;=&amp; \\displaystyle \\int_{x_1}^{x_2} \\left[ \\dfrac{\\partial f}{\\partial y(\\alpha, x)}  -  \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial \\frac{dy(\\alpha,x)}{dx}} \\right) \\right] \\eta(x) dx\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u56e0\u6b64\uff0c\u6839\u636e\u6761\u4ef6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\dfrac{\\partial J (x,y(\\alpha, x))}{\\partial \\alpha}\\right|_{\\alpha=0} = 0,<\/span><\/span> \u5e76\u4e14\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x)<\/span><\/span> \u662f\u6ee1\u8db3\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_2<\/span><\/span> \u5904\u4e3a\u96f6\u7684\u4efb\u610f\u51fd\u6570\uff0c\u56e0\u6b64\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\dfrac{\\partial f}{\\partial y(0, x)}  -  \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial \\frac{dy(0,x)}{dx}}\\right) = \\dfrac{\\partial f}{\\partial y(x)}  -  \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial \\frac{dy(x)}{dx}}\\right) = 0.\n\n<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u6700\u540e\uff0c\u901a\u8fc7\u7b80\u5316\u7b26\u53f7\uff0c\u6211\u4eec\u5f97\u5230\u8457\u540d\u7684\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\dfrac{\\partial f}{\\partial y}= \\dfrac{d}{dx}\\left( \\dfrac{\\partial f}{\\partial y^\\prime} \\right)},<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u8fd9\u8868\u793a\u4e86\u6cdb\u51fd <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u8fbe\u5230\u6781\u503c\u7684\u5fc5\u8981\u6761\u4ef6\u7684\u66f4\u7b80\u5355\u5f62\u5f0f\u3002<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u6700\u901f\u964d\u7ebf\u95ee\u9898<\/h2>\n<h3>\u95ee\u9898\u7684\u8868\u8ff0<\/h3>\n<p style=\"text-align:justify;\">\u6700\u901f\u964d\u7ebf\u95ee\u9898\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u673a\u68b0\u7269\u7406\u95ee\u9898\uff0c\u53ef\u4ee5\u901a\u8fc7\u53d8\u5206\u6cd5\u6765\u89e3\u51b3\u3002\u8bbe\u60f3\u4e00\u4e2a\u7269\u4f53\u5728\u6052\u5b9a\u529b\u573a\u7684\u4f5c\u7528\u4e0b\u4ece\u521d\u59cb\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)<\/span><\/span> \u79fb\u52a8\u5230\u7ec8\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span><\/span>\uff0c\u5176\u4e2d\u521d\u59cb\u70b9\u6bd4\u7ec8\u70b9\u4f4d\u7f6e\u9ad8\u3002\u95ee\u9898\u662f\uff1a\u7c92\u5b50\u5e94\u8be5\u9075\u5faa\u54ea\u6761\u8def\u5f84\u624d\u80fd\u4ee5\u6700\u77ed\u7684\u65f6\u95f4\u5230\u8fbe\u7ec8\u70b9\uff1f<\/p>\n<h3>\u95ee\u9898\u7684\u89e3\u51b3<\/h3>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u89e3\u51b3\u6700\u901f\u964d\u7ebf\u95ee\u9898\uff0c\u53ef\u4ee5\u7b80\u5355\u5316\u5730\u8003\u8651\u8be5\u60c5\u5f62\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u5c06\u8d77\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1)<\/span><\/span> \u8bbe\u5b9a\u5728\u5750\u6807\u539f\u70b9\uff0c\u800c\u7ec8\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span><\/span> \u4f4d\u4e8e\u539f\u70b9\u7684\u53f3\u4fa7\u5e76\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span> \u8f74\u4e4b\u4e0b\u3002<\/p>\n<div style=\"text-align:center;\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/probbraquistocrona.png\" alt=\"\u53d8\u5206\u6cd5 - \u6700\u901f\u964d\u7ebf\u95ee\u9898\" width=\"711\" height=\"505\" class=\"aligncenter size-full wp-image-28729 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/probbraquistocrona.png\" alt=\"\u53d8\u5206\u6cd5 - \u6700\u901f\u964d\u7ebf\u95ee\u9898\" width=\"711\" height=\"505\" class=\"aligncenter size-full wp-image-28729 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/probbraquistocrona.png 711w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/probbraquistocrona-300x213.png 300w\" sizes=\"(max-width: 711px) 100vw, 711px\" \/><\/noscript><\/div>\n<p style=\"text-align:justify;\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u8003\u8651\u7531\u91cd\u529b\u4ea7\u751f\u7684\u5411\u4e0b\u4f5c\u7528\u529b\u573a\uff08\u65b9\u5411\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\hat{y}<\/span><\/span>\uff09\uff0c\u5e76\u5047\u8bbe\u8fd0\u52a8\u5728\u6ca1\u6709\u6469\u64e6\u7684\u60c5\u51b5\u4e0b\u8fdb\u884c\u3002\u5728\u8fd9\u79cd\u80cc\u666f\u4e0b\uff0c\u7c92\u5b50\u88ab\u9650\u5236\u5728\u4e0d\u540c\u7684\u8def\u5f84\u4e0a\uff0c\u8fd9\u4e9b\u8def\u5f84\u8fde\u63a5\u51fa\u53d1\u70b9\u548c\u5230\u8fbe\u70b9\uff0c\u76ee\u7684\u662f\u627e\u5230\u54ea\u6761\u8def\u5f84\u53ef\u4ee5\u6700\u5c0f\u5316\u65c5\u884c\u65f6\u95f4\u3002<\/p>\n<h3>\u80fd\u91cf\u5206\u6790<\/h3>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u7cfb\u7edf\u7684\u80fd\u91cf\u5b88\u6052\u3002\u7cfb\u7edf\u7684\u603b\u80fd\u91cf\u4fdd\u6301\u4e0d\u53d8\uff0c\u8003\u8651\u5230\u52a8\u80fd <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{cin}=\\frac{1}{2}mv^2<\/span><\/span> \u548c\u91cd\u529b\u52bf\u80fd <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{pot,g}<\/span><\/span>\uff0c\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span> \u662f\u7c92\u5b50\u7684\u8d28\u91cf\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">v<\/span> \u662f\u5176\u901f\u5ea6\u3002\u5bf9\u4e8e\u52bf\u80fd\uff0c\u4ee5\u539f\u70b9\u4e3a\u53c2\u8003\uff0c\u56e0\u6b64 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{pot,g}(y=0)=0<\/span><\/span>\uff0c\u5728\u4efb\u4f55\u5176\u4ed6\u9ad8\u5ea6 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u5904\uff0c\u52bf\u80fd\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{pot,g}(y)=mgy.<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u7531\u4e8e\u7c92\u5b50\u4ece\u539f\u70b9\u51fa\u53d1\u4e14\u901f\u5ea6\u4e3a\u96f6\uff0c\u5176\u603b\u80fd\u91cf\u4e3a\u96f6\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{cin} + E_{pot,g}=0<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u7531\u4e8e\u7c92\u5b50\u4f4e\u4e8e\u53c2\u8003\u70b9\uff0c\u5b83\u7684\u52bf\u80fd\u5c06\u4e3a\u8d1f\uff0c\u52a8\u80fd\u5c06\u4e3a\u6b63\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4ece\u80fd\u91cf\u5b88\u6052\u65b9\u7a0b\u4e2d\u89e3\u51fa\u901f\u5ea6 <span class=\"katex-eq\" data-katex-display=\"false\">v<\/span>\uff1a<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{} &amp;\\dfrac{1}{2}mv^2 + (-mgy) = 0 \\\\ \\\\\n\n\\vdash &amp;\\dfrac{1}{2}mv^2 = mgy \\\\ \\\\\n\n\\vdash &amp;v^2 = 2gy \\\\ \\\\\n\n\\vdash &amp;v = \\sqrt{2gy}\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">\u8fd9\u6837\uff0c\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u7c92\u5b50\u6240\u5728\u8def\u5f84\u4e0a\u7684\u9ad8\u5ea6 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u6765\u8ba1\u7b97\u7c92\u5b50\u7684\u901f\u5ea6\u3002<\/p>\n<h3>\u884c\u7a0b\u65f6\u95f4\u5206\u6790<\/h3>\n<p style=\"text-align:justify;\">\u4e00\u65e6\u6211\u4eec\u5f97\u5230\u4e86\u79fb\u52a8\u7684\u901f\u5ea6\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528\u4f4d\u79fb\u5143\u7d20 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ds=\\sqrt{dx^2 + dy^2}<\/span><\/span> \u6765\u6784\u5efa\u884c\u7a0b\u65f6\u95f4\u5143\u7d20\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{} dt &amp;= \\dfrac{ds}{v} = \\dfrac{\\sqrt{dx^2 + dy^2}}{\\sqrt{2gy}}\\\\ \\\\\n\n&amp;= \\sqrt{\\dfrac{dx^2 + dy^2}{2gy} }\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u56e0\u6b64\uff0c\u4ece\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)<\/span><\/span> \u5230\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span><\/span> \u7684\u884c\u7a0b\u65f6\u95f4\u53ef\u4ee5\u901a\u8fc7\u79ef\u5206\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{} t &amp;= \\displaystyle \\int_{(x_1,y_1)}^{(x_2,y_2)} dt \\\\ \\\\\n\n&amp;= \\displaystyle \\int_{(x_1,y_1)}^{(x_2,y_2)} \\sqrt{\\dfrac{dx^2 + dy^2}{2gy}} \\\\ \\\\\n\n&amp;= \\displaystyle \\dfrac{1}{\\sqrt{2g}}\\int_{y_1}^{y_2} \\sqrt{\\dfrac{1+ \\left(\\dfrac{dx}{dy}\\right)^2 }{y}}dy \\\\ \\\\\n\n\\end{array}<\/span><\/span><\/p>\n<h3>\u53d8\u5206\u95ee\u9898\u7684\u8868\u8ff0<\/h3>\n<p style=\"text-align:justify;\">\u901a\u8fc7\u8fd9\u4e2a\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5df2\u7ecf\u5c06\u65f6\u95f4\u8868\u8fbe\u4e3a\u4e00\u4e2a\u6cdb\u51fd\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n{}t = J(y,x(y)) = \\displaystyle \\int_{y_1}^{y_2} f\\left(y,x(y),\\dfrac{dx(y)}{dy} \\right) dy\n\n<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u5176\u4e2d\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f\\left(y,x(y), \\dfrac{dx(y)}{dx}\\right) = \\sqrt{\\dfrac{1+ \\left(\\dfrac{dx(y)}{dy} \\right)^2}{y}} <\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u5728\u8fd9\u4e00\u70b9\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5ffd\u7565\u56e0\u5b50 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{2g},<\/span><\/span> \u56e0\u4e3a\u4f18\u5316 <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u548c\u4f18\u5316 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{2g}J<\/span><\/span> \u662f\u5b8c\u5168\u4e00\u6837\u7684\u3002<\/p>\n<p style=\"text-align:justify;\">\u6839\u636e\u4ee5\u4e0a\uff0c\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u6784\u5efa\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\uff0c\u4f7f\u7528\u4e4b\u524d\u7684\u65b9\u6cd5\uff0c\u6700\u7ec8\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\partial f}{\\partial x} = \\dfrac{d}{dy} \\dfrac{\\partial f}{\\partial x^\\prime}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\partial f}{\\partial x} = 0,<\/span><\/span> \u56e0\u6b64\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dy}\\dfrac{\\partial f}{\\partial x^\\prime} = 0,<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u6362\u53e5\u8bdd\u8bf4\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\partial f}{\\partial x^\\prime} = \\dfrac{1}{\\sqrt{2a}},<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u662f\u4e00\u4e2a\u4efb\u610f\u5e38\u6570\uff0c\u8fd9\u6837\u5199\u662f\u4e3a\u4e86\u65b9\u4fbf\u540e\u7eed\u7684\u63a8\u5bfc\u3002<\/p>\n<h3>\u53d8\u5206\u95ee\u9898\u7684\u89e3\u51b3<\/h3>\n<p style=\"text-align:justify;\">\u5c06\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u4ee3\u5165\u6700\u540e\u4e00\u4e2a\u8868\u8fbe\u5f0f\u4e2d\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{} &amp;\\dfrac{\\partial }{\\partial x^\\prime} \\sqrt{\\dfrac{1+ x^{\\prime 2}}{y}}  = \\dfrac{1}{\\sqrt{2a}} \\\\ \\\\\n\n\\vdash &amp; \\dfrac{1}{2}\\left( \\dfrac{1 + x^{\\prime 2} }{y} \\right)^{-1\/2} \\left(\\dfrac{2x^\\prime}{y} \\right) = \\dfrac{1}{\\sqrt{2a}} \\\\ \\\\\n\n\\vdash &amp; \\dfrac{1}{2}\\sqrt{\\dfrac{y}{1 + x^{\\prime 2}}} \\left(\\dfrac{2x^\\prime}{y} \\right) = \\dfrac{1}{\\sqrt{2a}} \\\\ \\\\\n\n\\vdash &amp; \\sqrt{\\dfrac{4x^{\\prime 2} y}{4y^2 (1 + x^{\\prime 2})} }  = \\sqrt{\\dfrac{1}{2a}} \\\\ \\\\\n\n\\vdash &amp;  \\dfrac{y x^{\\prime 2} }{y^2 (1 + x^{\\prime 2})}   = \\dfrac{1}{ 2a} \\\\ \\\\\n\n\\vdash &amp; 2ayx^{\\prime 2} = y^2 + y^2 x^{\\prime 2} \\\\ \\\\\n\n\\vdash &amp;  x^{\\prime 2} (2ay - y^2) = y^2 \\\\ \\\\\n\n\\vdash &amp; \\left(\\dfrac{dx}{dy}\\right)^2 = \\dfrac{y^2}{2ay - y^2} \\\\ \\\\\n\n\\vdash &amp; \\dfrac{dx}{dy} = \\pm \\sqrt{\\dfrac{y^2}{2ay - y^2}} \\\\ \\\\\n\n\\vdash &amp; dx = \\pm \\dfrac{ydy}{\\sqrt{2ay - y^2}} \\\\ \\\\\n\n\\vdash &amp; x = \\displaystyle  \\pm \\int \\dfrac{y}{\\sqrt{2ay - y^2}}dy\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u79ef\u5206\uff0c\u53ef\u4ee5\u8003\u8651\u8fdb\u884c\u4ee5\u4e0b\u66ff\u6362\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} y &amp;=&amp; a[1-\\cos(\\theta)] \\\\\n\n dy &amp;=&amp; a\\sin(\\theta) d\\theta\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u56e0\u6b64\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{} x= &amp; \\pm \\displaystyle \\int \\dfrac{y}{\\sqrt{2ay - y^2}}dy = \\displaystyle \\int \\dfrac{a[1-\\cos(\\theta)]a\\sin(\\theta)}{\\sqrt{2a^2[1-\\cos(\\theta)] - a^2[1-\\cos(\\theta)]^2 }}d\\theta \\\\ \\\\\n\n&amp; {} = \\pm \\displaystyle \\int \\dfrac{a^2[1-\\cos(\\theta)]\\sin(\\theta)}{\\sqrt{a^2[1-\\cos(\\theta)]\\left\\{ 2 - [1-\\cos(\\theta)] \\right\\} }}d\\theta \\\\ \\\\\n\n&amp; {} = \\pm \\displaystyle \\int \\dfrac{a[1-\\cos(\\theta)]\\sin(\\theta)}{\\sqrt{[1-\\cos(\\theta)]  [1 + \\cos(\\theta)]  }}d\\theta \\\\ \\\\\n\n&amp; {} = \\pm \\displaystyle \\int \\dfrac{a[1-\\cos(\\theta)]\\sin(\\theta)}{\\sqrt{ 1-\\cos^2(\\theta)}}d\\theta \\\\ \\\\\n\n&amp; {} = \\pm \\displaystyle \\int \\dfrac{a[1-\\cos(\\theta)]\\sin(\\theta)}{\\sin(\\theta)}d\\theta \\\\ \\\\\n\n&amp; {} = \\pm \\displaystyle \\int a[1-\\cos(\\theta)] d\\theta \\\\ \\\\\n\n&amp; {} = \\pm a(\\theta - \\sin(\\theta)) + C\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u6700\u901f\u964d\u7ebf\u66f2\u7ebf\u53ef\u4ee5\u8868\u793a\u4e3a\u6781\u5750\u6807\u4e2d\u7684\u53c2\u6570\u66f2\u7ebf\uff0c\u5b83\u4e0e\u4e00\u4e2a\u8d77\u70b9\u4f4d\u4e8e\u539f\u70b9\u7684\u6446\u7ebf\u4e00\u81f4\u3002<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} &amp; x(\\theta) &amp;=&amp; \\pm a(\\theta - \\sin(\\theta)) \\\\\n\n   &amp; y(\\theta) &amp;=&amp; a(1-\\cos(\\theta))\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u79ef\u5206\u5e38\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> \u4e3a\u96f6\u4ee5\u6ee1\u8db3\u8f68\u8ff9\u8d77\u70b9\u5728\u539f\u70b9\u7684\u521d\u59cb\u6761\u4ef6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u6709\u4e24\u7ec4\u65b9\u7a0b\u63d0\u4f9b\u4e86\u8be5\u95ee\u9898\u7684\u53ef\u80fd\u89e3\uff0c\u5176\u4e2d\u5e38\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u53ef\u4ee5\u8c03\u6574\uff0c\u4f7f\u66f2\u7ebf\u901a\u8fc7\u8def\u5f84\u7684\u7ec8\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span><\/span>\u3002\u8fd9\u4e9b\u65b9\u7a0b\u662f\uff1a<\/p>\n<p style=\"text-align:center;\"><strong>\u9009\u9879 1:<\/strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\begin{array}\n\n{} &amp; x(\\theta) &amp;=&amp; a(\\theta - \\sin(\\theta)) \\\\\n\n   &amp; y(\\theta) &amp;=&amp; a(1-\\cos(\\theta))\n\n\\end{array}}<\/span><\/span><\/p>\n<p style=\"text-align:center;\"><strong>\u9009\u9879 2:<\/strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\begin{array}\n\n{} &amp; x(\\theta) &amp;=&amp; - a(\\theta - \\sin(\\theta)) \\\\\n\n   &amp; y(\\theta) &amp;=&amp; a(1-\\cos(\\theta))\n\n\\end{array}}<\/span> <\/span><\/p>\n<p style=\"text-align:justify;\">\u6b64\u95ee\u9898\u7684\u53ef\u884c\u89e3\u7531\u7b2c\u4e8c\u4e2a\u9009\u9879\u7ed9\u51fa\uff0c\u901a\u8fc7\u5c06\u5e38\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u8bbe\u4e3a\u8d1f\u503c\uff0c\u6211\u4eec\u5f97\u5230\u4e00\u6761\u7b26\u5408\u6210\u4e3a\u89e3\u7684\u5fc5\u8981\u6761\u4ef6\u7684\u66f2\u7ebf\u3002<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/solbraquis.png\" alt=\"\u53ef\u80fd\u7684\u89e3\u793a\u4f8b\uff0c\u4e00\u4e2a\u6446\u7ebf\u5f27\" width=\"497\" height=\"329\" class=\"aligncenter size-full wp-image-28731 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/solbraquis.png\" alt=\"\u53ef\u80fd\u7684\u89e3\u793a\u4f8b\uff0c\u4e00\u4e2a\u6446\u7ebf\u5f27\" width=\"497\" height=\"329\" class=\"aligncenter size-full wp-image-28731 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/solbraquis.png 497w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/solbraquis-300x199.png 300w\" sizes=\"(max-width: 497px) 100vw, 497px\" \/><\/noscript><\/center><\/p>\n<h3>\u89e3\u7684\u6700\u7ec8\u8c03\u6574<\/h3>\n<p style=\"text-align:justify;\">\u7ecf\u8fc7\u6700\u540e\u7684\u8c03\u6574\uff0c\u6700\u901f\u964d\u7ebf\u66f2\u7ebf\u5177\u6709\u4ee5\u4e0b\u53c2\u6570\u5f62\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} x(\\theta) &amp;= b(\\theta - \\sin(\\theta)) \\\\\n\n   y(\\theta) &amp;= -b(1-\\cos(\\theta))\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align:justify;\">\u6211\u4eec\u66ff\u6362\u4e86 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=-b,<\/span><\/span> \u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt b.<\/span><\/span> \u66f2\u7ebf\u7684\u5468\u671f\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2b\\pi<\/span><\/span>\uff0c\u5fc5\u987b\u6ee1\u8db3\u6761\u4ef6 <span class=\"katex-eq\" data-katex-display=\"false\">x_2 \\in ]0,2b\\pi[<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">y_2 \\in ]-2b,0[.<\/span> \u8fd9\u6700\u540e\u4e00\u70b9\u5f88\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u8981\u6c42\u6700\u901f\u964d\u7ebf\u66f2\u7ebf\u8868\u793a\u4e3a\u5355\u6446\u7ebf\u5f27\uff0c\u5982\u679c\u7c92\u5b50\u5728\u8fd4\u56de\u96f6\u9ad8\u5ea6\u70b9\u65f6\u505c\u6b62\uff0c\u89e3\u5c06\u4e0d\u518d\u6709\u6548\u3002<\/p>\n<p style=\"text-align:justify;\">\u4e3a\u4e86\u5c06\u8fd9\u4e9b\u65b9\u7a0b\u8c03\u6574\u5230\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u627e\u5230\u6ee1\u8db3\u4ee5\u4e0b\u7cfb\u7edf\u7684 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u7684\u503c\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} x_2 &amp;= b(\\theta - \\sin(\\theta))\\\\\n\ny_2 &amp;= - b(1-\\cos(\\theta))\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">\u8fd9\u4e2a\u975e\u7ebf\u6027\u7cfb\u7edf\u4f3c\u4e4e\u6ca1\u6709\u89e3\u6790\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u5728 Wolfram Mathematica \u4e2d\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\u3002\u4e0b\u9762\u662f\u89e3\u51b3\u95ee\u9898\u7684\u4e00\u7cfb\u5217\u6b65\u9aa4\uff1a<\/p>\n<p><\/br><\/p>\n<h4>\u6b65\u9aa4 1: \u5efa\u7acb\u7cfb\u7edf<\/h4>\n<p style=\"text-align:justify;\">\u5efa\u7acb\u9700\u8981\u89e3\u51b3\u7684\u7cfb\u7edf\u65b9\u7a0b<\/p>\n<p><span dir=\"ltr\"><code>eq1 = x2 == b*(theta - Sin[theta])<br \/>\n eq2 = y2 == -b*(1 - Cos[theta])<\/code><\/span><br \/>\n<\/br><\/p>\n<h4>\u6b65\u9aa4 2: \u5b9a\u4e49\u5230\u8fbe\u70b9<\/h4>\n<p style=\"text-align:justify;\">\u5b9a\u4e49\u7c92\u5b50\u5230\u8fbe\u8def\u5f84\u7ec8\u70b9\u7684\u4f4d\u7f6e\u3002\u5728\u6b64\u6848\u4f8b\u4e2d\uff0c\u6211\u4eec\u5c06\u5176\u8bbe\u7f6e\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)=(1,-2).<\/span><\/span> \u4f60\u53ef\u4ee5\u4fee\u6539\u8fd9\u4e9b\u503c\u4ee5\u6d4b\u8bd5\u5176\u4ed6\u7c7b\u4f3c\u7684\u914d\u7f6e\u3002<\/p>\n<p><span dir=\"ltr\"><code>x2val = 1; y2val = -2;<\/code><\/span><br \/>\n<\/br><\/p>\n<h4>\u6b65\u9aa4 3: \u6570\u503c\u8ba1\u7b97\u6240\u9700\u7684\u503c<\/h4>\n<p style=\"text-align:justify;\">\u4f7f\u7528 \u00abFindRoot\u00bb \u51fd\u6570\u6765\u6570\u503c\u8ba1\u7b97\u95ee\u9898\u7684\u89e3<\/p>\n<p><span dir=\"ltr\"><code>sol = FindRoot[{eq1, eq2} \/. {x2 -> x2val, y2 -> y2val}, {{b,1}, {theta, 1}}]<\/code><\/span><\/p>\n<p style=\"text-align:justify;\">\u8fd9\u91cc\u4f7f\u7528\u4e86 <span class=\"katex-eq\" data-katex-display=\"false\">b=1<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\theta=1<\/span> \u4f5c\u4e3a\u6570\u503c\u89e3\u7684\u521d\u59cb\u8fd1\u4f3c\u70b9\u3002\u5f97\u5230\u7684\u89e3\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\approx 2.4056<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta \\approx 1.40138<\/span><\/span><\/p>\n<p><\/br><\/p>\n<h4>\u6b65\u9aa4 4: \u7ed3\u679c\u9a8c\u8bc1<\/h4>\n<p style=\"text-align:justify;\">\u6211\u4eec\u8bb0\u4f4f\uff0c\u4e3a\u4e86\u8fd9\u4e9b\u7b54\u6848\u5177\u6709\u7269\u7406\u610f\u4e49\uff0c\u5fc5\u987b\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_2 \\in ]0,2b\\pi[<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y_2 \\in ]-2b, 0[.<\/span><\/span> \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u5feb\u901f\u65b9\u6cd5\u9a8c\u8bc1\u8fd9\u4e00\u70b9<\/p>\n<p style=\"text-align:justify;\">\u9996\u5148\u63d0\u53d6\u89e3\u4e2d\u5f97\u5230\u7684 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> \u7684\u503c<\/p>\n<p><code>bval = sol[[1, 2]]; thetaval = sol[[2, 2]];<\/code><\/p>\n<p style=\"text-align:justify;\">\u7136\u540e\u6267\u884c\u786e\u8ba4<\/p>\n<p><code>If[0 < x2val < 2*Pi*bval &#038;&#038; -2*bval < y2val < 0 \"\u6709\u6548\u503c\", \"\u65e0\u6548\u503c\"]\n<\/code><\/p>\n<p style=\"text-align:justify;\">\u5982\u679c\u4e00\u5207\u987a\u5229\uff0c\u6211\u4eec\u5e94\u8be5\u5728\u8f93\u51fa\u4e2d\u5f97\u5230 \"\u6709\u6548\u503c\"\u3002\u8fd9\u6bb5\u4ee3\u7801\u5c06\u5e2e\u52a9\u4f60\u68c0\u67e5\u7269\u7406\u60c5\u51b5\u662f\u5426\u6b63\u786e\u5efa\u6a21\u3002<\/p>\n<p style=\"text-align:justify;\">\u901a\u8fc7\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u6211\u4eec\u6700\u7ec8\u5b8c\u5168\u8c03\u6574\u4e86\u6211\u4eec\u7684\u89e3\u66f2\u7ebf\uff0c\u5b83\u8fde\u63a5\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)=(0,0)<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)=(1,-2).<\/span><\/span> \u6700\u7ec8\u66f2\u7ebf\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} x(\\theta) &amp;\\approx 2.4056(\\theta - \\sin(\\theta)) \\\\\n\ny(\\theta) &amp;\\approx -2.4056(1-\\cos(\\theta))\n\n\\end{array}\\;\\;;\\theta\\in [0, 1.40138]<\/span>\n<p style=\"text-align:justify;\">\u56fe\u5f62\u4e0a\u663e\u793a\u5982\u4e0b\uff1a<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/sol2braquis.png\" alt=\"\" width=\"296\" height=\"345\" class=\"aligncenter size-full wp-image-28733 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/sol2braquis.png\" alt=\"\" width=\"296\" height=\"345\" class=\"aligncenter size-full wp-image-28733 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/sol2braquis.png 296w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/sol2braquis-257x300.png 257w\" sizes=\"(max-width: 296px) 100vw, 296px\" \/><\/noscript><\/center><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u5e26\u6709Wolfram\u7b97\u6cd5\u7684Github\u4ed3\u5e93<\/h2>\n<p style=\"text-align:justify;\">\u89e3\u51b3\u6700\u901f\u964d\u7ebf\u95ee\u9898\u7684\u5b8c\u6574\u4ee3\u7801\uff0c\u5305\u62ec\u5728Wolfram Mathematica\u4e2d\u5f00\u53d1\u7684\u7b97\u6cd5\uff0c\u53ef\u5728\u6211\u7684GitHub\u4ed3\u5e93\u4e2d\u4e0b\u8f7d\u548c\u67e5\u770b\u3002\u8be5\u4ed3\u5e93\u5305\u62ec\u4e00\u4e2a\u5305\u542b\u4ea4\u4e92\u5f0f\u7b14\u8bb0\u672c\u683c\u5f0f\u4ee3\u7801\u7684`.nb`\u6587\u4ef6\uff0c\u4ee5\u53ca\u4e00\u4e2a\u7eaf\u6587\u672c\u7248\u672c\u7684`.m`\u6587\u4ef6\uff0c\u9002\u7528\u4e8e\u90a3\u4e9b\u66f4\u559c\u6b22\u76f4\u63a5\u67e5\u770b\u4ee3\u7801\u7684\u4eba\u3002<\/p>\n<p style=\"text-align:justify;\"><strong>\u4f60\u53ef\u4ee5\u4eceGitHub<a href=\"https:\/\/github.com\/girebz\/Braquist-crona\" target=\"_blank\" rel=\"noopener\">\u4e0b\u8f7d\u4ed3\u5e93<\/a>\u3002<\/strong><\/p>\n<p style=\"text-align:justify;\">\u9664\u4e86\u4ee3\u7801\uff0c\u4ed3\u5e93\u8fd8\u5305\u542b\u4e00\u4e2a\"README\"\u6587\u4ef6\uff0c\u5176\u4e2d\u8be6\u7ec6\u8bf4\u660e\u4e86\u5982\u4f55\u4f7f\u7528\u548c\u7406\u89e3\u7b97\u6cd5\uff0c\u5e76\u63d0\u4f9b\u4e86\u9010\u6b65\u89e3\u51b3\u6700\u901f\u964d\u7ebf\u95ee\u9898\u7684\u89e3\u91ca\u3002\u5e0c\u671b\u4f60\u89c9\u5f97\u5b83\u6709\u7528\uff01<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u7ecf\u5178\u529b\u5b66\u4e2d\u7684\u53d8\u5206\u6cd5\u4e0e\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b \u6458\u8981\uff1a\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u53d8\u5206\u6cd5\u6280\u672f\u63a8\u5bfc\u51fa\u5206\u6790\u529b\u5b66\u4e2d\u7684\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\uff0c\u5e76\u8be6\u7ec6\u5c55\u793a\u5176\u5728\u89e3\u51b3\u6700\u901f\u964d\u7ebf\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002 \u5b66\u4e60\u76ee\u6807\uff1a 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