{"id":28789,"date":"2024-09-19T17:27:47","date_gmt":"2024-09-19T17:27:47","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28789"},"modified":"2024-09-19T17:39:11","modified_gmt":"2024-09-19T17:39:11","slug":"%d0%b1%d1%80%d0%b0%d1%85%d0%b8%d1%81%d1%82%d0%be%d1%85%d1%80%d0%be%d0%bd%d0%b0-%d0%b8-%d1%83%d1%80%d0%b0%d0%b2%d0%bd%d0%b5%d0%bd%d0%b8%d0%b5-%d1%8d%d0%b9%d0%bb%d0%b5%d1%80%d0%b0-%d0%bb%d0%b0%d0%b3","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d0%b1%d1%80%d0%b0%d1%85%d0%b8%d1%81%d1%82%d0%be%d1%85%d1%80%d0%be%d0%bd%d0%b0-%d0%b8-%d1%83%d1%80%d0%b0%d0%b2%d0%bd%d0%b5%d0%bd%d0%b8%d0%b5-%d1%8d%d0%b9%d0%bb%d0%b5%d1%80%d0%b0-%d0%bb%d0%b0%d0%b3\/","title":{"rendered":"\u0411\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0430 \u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 \u0441 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href=\"#2\">\u0424\u041e\u0420\u041c\u0423\u041b\u0418\u0420\u041e\u0412\u041a\u0410 \u0412\u0410\u0420\u0418\u0410\u0426\u0418\u041e\u041d\u041d\u041e\u0419 \u0417\u0410\u0414\u0410\u0427\u0418<\/a><br \/>\n<a href=\"#3\">\u0423\u0420\u0410\u041d\u0415\u041d\u0418\u0415 \u042d\u0419\u041b\u0415\u0420\u0410-\u041b\u0410\u0413\u0420\u0410\u041d\u0416\u0410<\/a><br \/>\n<a href=\"#4\">\u041f\u0420\u041e\u0411\u041b\u0415\u041c\u0410 \u0411\u0420\u0410\u0425\u0418\u0421\u0422\u041e\u0425\u0420\u041e\u041d\u042b<\/a><br \/>\n<a href=\"#5\">\u0420\u0415\u041f\u041e\u0417\u0418\u0422\u041e\u0420\u0418\u0419 GITHUB \u0421 \u0410\u041b\u0413\u041e\u0420\u0418\u0422\u041c\u041e\u041c WOLFRAM<\/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\" 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\u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0442\u043e\u0447\u043d\u044b\u0435 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u0441\u0438\u0441\u0442\u0435\u043c \u0441 \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0435\u0439 \u0438 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u043d\u044b\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0442\u0435\u043e\u0440\u0438\u0438 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u0432\u043e\u0437\u043c\u0443\u0449\u0435\u043d\u0438\u0439 \u0434\u043b\u044f \u0431\u043e\u043b\u0435\u0435 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u0441\u0438\u0441\u0442\u0435\u043c. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0432 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044f\u0445, \u043a\u043e\u0433\u0434\u0430 \u0441\u043b\u043e\u0436\u043d\u043e \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u0441\u0438\u043b\u044b \u0432 \u0442\u0435\u0440\u043c\u0438\u043d\u0430\u0445 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u044b\u0445 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439, \u043f\u0440\u0438\u043d\u0446\u0438\u043f \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0435\u0433\u043e \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0431\u043e\u043b\u0435\u0435 \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u044b\u0439 \u043c\u0435\u0442\u043e\u0434 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0438. \u0412 \u043e\u0431\u0449\u0435\u043c, \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0435 \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u2014 \u044d\u0442\u043e \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u0439 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u0442 \u0430\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u0443\u044e \u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u043e\u0432\u043a\u0443 \u0437\u0430\u043a\u043e\u043d\u043e\u0432 \u041d\u044c\u044e\u0442\u043e\u043d\u0430, \u0443\u043d\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044e \u0437\u0430\u043a\u043e\u043d\u043e\u0432 \u0444\u0438\u0437\u0438\u043a\u0438, \u043f\u043e\u0432\u044b\u0448\u0435\u043d\u043d\u0443\u044e \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u044c \u0432 \u0440\u0435\u0448\u0435\u043d\u0438\u0438 \u0437\u0430\u0434\u0430\u0447 \u0438 \u0431\u043e\u043b\u044c\u0448\u0443\u044e \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c \u0432 \u043f\u0440\u0435\u0434\u0441\u043a\u0430\u0437\u0430\u043d\u0438\u0438 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u0445 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0424\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u043e\u0432\u043a\u0430 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438<\/h2>\n<p style=\"text-align:justify;\">\u0412\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0435 \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u043e \u043d\u0430 \u043f\u043e\u0438\u0441\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span>, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u0442 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u0430:<\/p>\n<p style=\"text-align:center;\"><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>\n<p style=\"text-align:justify;\">\u0441 \u0446\u0435\u043b\u044c\u044e \u043d\u0430\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u0435\u0433\u043e \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u0438\u043b\u0438 \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u0430. \u0412 \u044d\u0442\u043e\u043c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span> \u0438 \u0435\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 <span class=\"katex-eq\" data-katex-display=\"false\">dy(x)\/dx,<\/span> \u0432 \u0442\u043e \u0432\u0440\u0435\u043c\u044f \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u043e\u0441\u0442\u0430\u044e\u0442\u0441\u044f \u0444\u0438\u043a\u0441\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u044b\u043c\u0438. \u0427\u0442\u043e\u0431\u044b \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b, \u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span> \u043d\u0430\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u044e\u0442\u0441\u044f \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u0438, \u0441 \u0446\u0435\u043b\u044c\u044e \u043d\u0430\u0439\u0442\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044e, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0434\u0435\u043b\u0430\u0435\u0442 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u0430 \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u044c\u043d\u044b\u043c. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0435\u0441\u043b\u0438 \u0443\u0434\u0430\u0435\u0442\u0441\u044f \u0434\u043e\u0431\u0438\u0442\u044c\u0441\u044f \u043c\u0438\u043d\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0430, \u043b\u044e\u0431\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f <em>\u0432 \u0435\u0433\u043e \u043e\u043a\u0440\u0435\u0441\u0442\u043d\u043e\u0441\u0442\u0438<\/em>, \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e \u043e\u0442 \u0442\u043e\u0433\u043e, \u043d\u0430\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u043e\u043d\u0430 \u0431\u043b\u0438\u0437\u043a\u0430 \u043a <span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span>, \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u0442 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u0430.<\/p>\n<p style=\"text-align:justify;\">\u0414\u043b\u044f \u0432\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043f\u043e\u043d\u044f\u0442\u0438\u044f <em>\u0441\u043e\u0441\u0435\u0434\u043d\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/em> \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u0440\u0438\u0441\u0432\u043e\u0438\u0442\u044c \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">y=(\\alpha,x)<\/span> \u0432\u0441\u0435\u043c \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u044b\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044f\u043c <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0435\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=0<\/span>, \u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">y(0,x)=y(x)<\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0435\u0439, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u0442 <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span>. \u042d\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y(\\alpha, x) = y(x) + \\alpha \\eta(x),<\/span>\n<p style=\"text-align:justify;\">\u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x)<\/span> \u2014 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u043b\u0430\u0441\u0441\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}^1<\/span>, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">x_1<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">x_2<\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">y(\\alpha,x)<\/span>, \u0432\u043a\u043b\u044e\u0447\u0430\u044e\u0449\u0430\u044f \u044d\u0442\u043e \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u044e, \u0438\u0434\u0435\u043d\u0442\u0438\u0447\u043d\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span> \u0432 \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u044b\u0445 \u0438 \u043a\u043e\u043d\u0435\u0447\u043d\u044b\u0445 \u0442\u043e\u0447\u043a\u0430\u0445 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438 \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f.<\/p>\n<p style=\"text-align:justify;\">\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0432 \u0444\u0443\u043d\u043a\u0446\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">y(\\alpha,x)<\/span>, \u0432\u043a\u043b\u044e\u0447\u0430\u044e\u0449\u0443\u044e \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x)<\/span> \u0432\u043c\u0435\u0441\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span> \u0432 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0449\u0438\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span>, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u043d\u043e\u0432\u044b\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b, \u0437\u0430\u0432\u0438\u0441\u044f\u0449\u0438\u0439 \u043e\u0442 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span>:<\/p>\n<p style=\"text-align:center;\"><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>\n<p style=\"text-align:justify;\">\u0414\u043b\u044f \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043d\u0438\u044f \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u044b\u0445 \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u043e\u0432 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u044f:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\dfrac{\\partial J(x,y(\\alpha,x))}{\\partial \\alpha}\\right|_{\\alpha=0} = 0<\/span>\n<p style=\"text-align:justify;\">\u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x).<\/span>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0423\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430<\/h2>\n<p style=\"text-align:justify;\">\u0410\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0443\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e <span class=\"katex-eq\" data-katex-display=\"false\">\\partial J(x,y(\\alpha,x))\/\\partial \\alpha<\/span>, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/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;\">\u0421 \u044d\u0442\u043e\u0433\u043e \u043c\u043e\u043c\u0435\u043d\u0442\u0430 \u0432\u0430\u0436\u043d\u043e \u0437\u0430\u043c\u0435\u0442\u0438\u0442\u044c, \u0447\u0442\u043e:<\/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;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0441\u043e\u043a\u0440\u0430\u0449\u0430\u0435\u0442\u0441\u044f, \u043a\u0430\u043a \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u043e \u043d\u0438\u0436\u0435:<\/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;\">\u0415\u0441\u043b\u0438 \u0437\u0430\u0442\u0435\u043c \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0432\u0442\u043e\u0440\u043e\u0439 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b, \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c, \u0447\u0442\u043e \u0435\u0433\u043e \u043c\u043e\u0436\u043d\u043e \u0443\u043f\u0440\u043e\u0441\u0442\u0438\u0442\u044c, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 \u043f\u043e \u0447\u0430\u0441\u0442\u044f\u043c:<\/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;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c<\/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;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\dfrac{\\partial J (x,y(\\alpha, x))}{\\partial \\alpha}\\right|_{\\alpha=0} = 0,<\/span> \u0438 \u0442\u0430\u043a \u043a\u0430\u043a <span class=\"katex-eq\" data-katex-display=\"false\">\\eta(x)<\/span> &#8211; \u044d\u0442\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u043f\u043e\u0434\u0447\u0438\u043d\u0435\u043d\u043d\u0430\u044f \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u043c\u0443 \u0443\u0441\u043b\u043e\u0432\u0438\u044e \u0431\u044b\u0442\u044c \u0440\u0430\u0432\u043d\u043e\u0439 \u043d\u0443\u043b\u044e \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">x_1<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">x_2,<\/span>, \u0438\u043c\u0435\u0435\u043c:<\/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;\">\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u00ab\u0441\u0431\u0440\u043e\u0441\u0438\u0432 \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\u00bb \u0432 \u044d\u0442\u043e\u043c \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0438, \u043c\u044b \u043f\u0440\u0438\u0445\u043e\u0434\u0438\u043c \u043a \u0442\u043e\u043c\u0443, \u0447\u0442\u043e \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u043e \u043a\u0430\u043a \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430:<\/p>\n<p style=\"text-align:center;\"><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>\n<p style=\"text-align:justify;\">\u0438 \u044d\u0442\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0433\u043e\u0440\u0430\u0437\u0434\u043e \u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u043c \u0441\u043f\u043e\u0441\u043e\u0431\u043e\u043c \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u0434\u043b\u044f \u0442\u043e\u0433\u043e, \u0447\u0442\u043e\u0431\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u0434\u043e\u0441\u0442\u0438\u0433\u0430\u043b \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u0417\u0430\u0434\u0430\u0447\u0430 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435<\/h2>\n<h3>\u041f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430 \u0437\u0430\u0434\u0430\u0447\u0438<\/h3>\n<p style=\"text-align:justify;\">\u0417\u0430\u0434\u0430\u0447\u0430 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435 \u2014 \u044d\u0442\u043e \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0430 \u043c\u0435\u0445\u0430\u043d\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0444\u0438\u0437\u0438\u043a\u0438, \u0440\u0435\u0448\u0430\u0435\u043c\u0430\u044f \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0433\u043e \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f. \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0443\u044e \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044e: \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u0443 \u043d\u0430\u0441 \u0435\u0441\u0442\u044c \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u044c\u043d\u044b\u0439 \u043e\u0431\u044a\u0435\u043a\u0442, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0434\u0432\u0438\u0436\u0435\u0442\u0441\u044f \u043f\u043e\u0434 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0435\u043c \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e\u0433\u043e \u0441\u0438\u043b\u043e\u0432\u043e\u0433\u043e \u043f\u043e\u043b\u044f \u0438 \u043f\u0435\u0440\u0435\u043c\u0435\u0449\u0430\u0435\u0442\u0441\u044f \u0438\u0437 \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)<\/span> \u0432 \u043a\u043e\u043d\u0435\u0447\u043d\u0443\u044e \u0442\u043e\u0447\u043a\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span>, \u0433\u0434\u0435 \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u0430\u044f \u0442\u043e\u0447\u043a\u0430 \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u0432\u044b\u0448\u0435 \u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0439. \u0412\u043e\u043f\u0440\u043e\u0441 \u0437\u0430\u043a\u043b\u044e\u0447\u0430\u0435\u0442\u0441\u044f \u0432 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c: \u043a\u0430\u043a\u043e\u0439 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0435\u0439 \u0434\u043e\u043b\u0436\u043d\u0430 \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u044c \u0447\u0430\u0441\u0442\u0438\u0446\u0430, \u0447\u0442\u043e\u0431\u044b \u0434\u043e\u0441\u0442\u0438\u0447\u044c \u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0438 \u0437\u0430 \u043c\u0438\u043d\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0432\u0440\u0435\u043c\u044f?<\/p>\n<h3>\u041f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f<\/h3>\n<p style=\"text-align:justify;\">\u0414\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435 \u043f\u043e\u043b\u0435\u0437\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044e \u0432 \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u043d\u043e\u0439 \u0444\u043e\u0440\u043c\u0435. \u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043c\u043e\u0436\u043d\u043e \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u044c \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u0443\u044e \u0442\u043e\u0447\u043a\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1)<\/span> \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442, \u0430 \u043a\u043e\u043d\u0435\u0447\u043d\u0430\u044f \u0442\u043e\u0447\u043a\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span> \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u0441\u043f\u0440\u0430\u0432\u0430 \u043e\u0442 \u043d\u0430\u0447\u0430\u043b\u0430 \u0438 \u043d\u0438\u0436\u0435 \u043e\u0441\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span>.<\/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=\"\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0435 \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 - \u0437\u0430\u0434\u0430\u0447\u0430 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435\" 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=\"\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0435 \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 - \u0437\u0430\u0434\u0430\u0447\u0430 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435\" width=\"711\" height=\"505\" class=\"aligncenter size-full wp-image-28729 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/probbraquistocrona.png 711w, https:\/\/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;\">\u0412 \u044d\u0442\u043e\u0439 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u0438 \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0441\u0438\u043b\u043e\u0432\u043e\u0435 \u043f\u043e\u043b\u0435, \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u044e\u0449\u0435\u0435 \u0432\u043d\u0438\u0437 (\u0432 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">-\\hat{y}<\/span>), \u0441\u043e\u0437\u0434\u0430\u0432\u0430\u0435\u043c\u043e\u0435 \u0433\u0440\u0430\u0432\u0438\u0442\u0430\u0446\u0438\u0435\u0439, \u0438 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u0442\u044c, \u0447\u0442\u043e \u0434\u0432\u0438\u0436\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0431\u0435\u0437 \u0442\u0440\u0435\u043d\u0438\u044f. \u0412 \u044d\u0442\u043e\u043c \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u0447\u0430\u0441\u0442\u0438\u0446\u0430 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u044b\u043c\u0438 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u044f\u043c\u0438, \u0441\u043e\u0435\u0434\u0438\u043d\u044f\u044e\u0449\u0438\u043c\u0438 \u0442\u043e\u0447\u043a\u0438 \u0432\u044b\u0445\u043e\u0434\u0430 \u0438 \u043f\u0440\u0438\u0431\u044b\u0442\u0438\u044f, \u0441 \u0446\u0435\u043b\u044c\u044e \u043d\u0430\u0439\u0442\u0438 \u0442\u0443, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u0442 \u0432\u0440\u0435\u043c\u044f \u043f\u0443\u0442\u0435\u0448\u0435\u0441\u0442\u0432\u0438\u044f.<\/p>\n<h3>\u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0435 \u044d\u043d\u0435\u0440\u0433\u0438\u0438<\/h3>\n<p style=\"text-align:justify;\">\u0414\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u044d\u0442\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c\u0441\u044f \u0437\u0430\u043a\u043e\u043d\u043e\u043c \u0441\u043e\u0445\u0440\u0430\u043d\u0435\u043d\u0438\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u0438 \u0433\u0440\u0430\u0432\u0438\u0442\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u0441\u0438\u0441\u0442\u0435\u043c\u044b. \u041f\u043e\u043b\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u044b \u0431\u0443\u0434\u0435\u0442 \u043e\u0441\u0442\u0430\u0432\u0430\u0442\u044c\u0441\u044f \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e\u0439, \u0443\u0447\u0438\u0442\u044b\u0432\u0430\u044f \u043a\u0430\u043a \u043a\u0438\u043d\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u044d\u043d\u0435\u0440\u0433\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">E_{cin}=\\frac{1}{2}mv^2<\/span>, \u0442\u0430\u043a \u0438 \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u0443\u044e \u0433\u0440\u0430\u0432\u0438\u0442\u0430\u0446\u0438\u043e\u043d\u043d\u0443\u044e \u044d\u043d\u0435\u0440\u0433\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">E_{pot,g}<\/span>, \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span> \u2014 \u043c\u0430\u0441\u0441\u0430 \u0447\u0430\u0441\u0442\u0438\u0446\u044b, \u0430 <span class=\"katex-eq\" data-katex-display=\"false\">v<\/span> \u2014 \u0435\u0435 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c. \u0414\u043b\u044f \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0439 \u044d\u043d\u0435\u0440\u0433\u0438\u0438 \u0431\u044b\u043b \u0432\u0437\u044f\u0442 \u0437\u0430 \u043e\u0441\u043d\u043e\u0432\u0443 \u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a, \u0442\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">E_{pot,g}(y=0)=0<\/span>, \u0442\u043e\u0433\u0434\u0430 \u043a\u0430\u043a \u043d\u0430 \u043b\u044e\u0431\u043e\u0439 \u0434\u0440\u0443\u0433\u043e\u0439 \u0432\u044b\u0441\u043e\u0442\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u0438\u043c\u0435\u0435\u043c <span class=\"katex-eq\" data-katex-display=\"false\">E_{pot,g}(y)=mgy.<\/span>\n<p style=\"text-align:justify;\">\u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0447\u0430\u0441\u0442\u0438\u0446\u0430 \u043d\u0430\u0447\u0438\u043d\u0430\u0435\u0442 \u0434\u0432\u0438\u0436\u0435\u043d\u0438\u0435 \u0438\u0437 \u043f\u043e\u043a\u043e\u044f, \u0435\u0435 \u043f\u043e\u043b\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e. \u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0438\u043c\u0435\u0435\u043c:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{cin} + E_{pot,g}=0<\/span>\n<p style=\"text-align:justify;\">\u0422\u0430\u043a \u043a\u0430\u043a \u0447\u0430\u0441\u0442\u0438\u0446\u0430 \u043f\u0430\u0434\u0430\u0435\u0442 \u043d\u0438\u0436\u0435 \u0442\u043e\u0447\u043a\u0438 \u043e\u0442\u0441\u0447\u0435\u0442\u0430, \u0435\u0435 \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u0431\u0443\u0434\u0435\u0442 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439, \u0430 \u043a\u0438\u043d\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u2014 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439. \u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c <span class=\"katex-eq\" data-katex-display=\"false\">v<\/span> \u0438\u0437 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0441\u043e\u0445\u0440\u0430\u043d\u0435\u043d\u0438\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u0438 \u0438 \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c:<\/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;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0447\u0430\u0441\u0442\u0438\u0446\u044b \u0432 \u043b\u044e\u0431\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0435\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438 \u0432 \u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442\u0438 \u043e\u0442 \u0432\u044b\u0441\u043e\u0442\u044b <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u043e\u043d\u0430 \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f.<\/p>\n<h3>\u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0435 \u0432\u0440\u0435\u043c\u0435\u043d\u0438 \u043f\u0440\u043e\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f<\/h3>\n<p style=\"text-align:justify;\">\u041a\u0430\u043a \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0434\u0432\u0438\u0436\u0435\u043d\u0438\u044f, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u0432\u0440\u0435\u043c\u0435\u043d\u0438 \u043f\u0440\u043e\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u0441\u043c\u0435\u0449\u0435\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">ds=\\sqrt{dx^2 + dy^2}<\/span> \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/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;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0432\u0440\u0435\u043c\u044f \u043f\u0440\u043e\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0442\u043e\u0447\u043a\u0430\u043c\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span> \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c, \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u044f<\/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>\u041f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438<\/h3>\n<p style=\"text-align:justify;\">\u0421 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u044d\u0442\u043e\u0433\u043e \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u0433\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u043c\u044b \u0441\u043c\u043e\u0433\u043b\u0438 \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u0432\u0440\u0435\u043c\u044f \u043a\u0430\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b \u0432 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0444\u043e\u0440\u043c\u0435<\/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;\">\u0433\u0434\u0435<\/p>\n<p style=\"text-align:center;\"><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>\n<p style=\"text-align:justify;\">\u041d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043e\u043f\u0443\u0441\u0442\u0438\u0442\u044c \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c <span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{2g},<\/span> \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">J<\/span> \u0438\u0434\u0435\u043d\u0442\u0438\u0447\u043d\u0430 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{2g}J<\/span>.<\/p>\n<p style=\"text-align:justify;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0442\u0435\u043f\u0435\u0440\u044c \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430, \u0441\u043b\u0435\u0434\u0443\u044f \u0442\u043e\u0439 \u0436\u0435 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u0435, \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043c\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0438 \u0440\u0430\u043d\u0435\u0435, \u0438 \u0432 \u0438\u0442\u043e\u0433\u0435 \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\partial f}{\\partial x} = \\dfrac{d}{dy} \\dfrac{\\partial f}{\\partial x^\\prime}<\/span>\n<p style=\"text-align:justify;\">\u041e\u0434\u043d\u0430\u043a\u043e \u0437\u0434\u0435\u0441\u044c \u043c\u044b \u0432\u0438\u0434\u0438\u043c, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\partial f}{\\partial x} = 0,<\/span>, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0438\u043c\u0435\u0435\u043c<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dy}\\dfrac{\\partial f}{\\partial x^\\prime} = 0,<\/span>\n<p style=\"text-align:justify;\">\u0438\u043b\u0438, \u0434\u0440\u0443\u0433\u0438\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\partial f}{\\partial x^\\prime} = \\dfrac{1}{\\sqrt{2a}},<\/span>\n<p style=\"text-align:justify;\">\u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u2014 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u0430\u044f \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u0430\u044f, \u0437\u0430\u043f\u0438\u0441\u0430\u043d\u043d\u0430\u044f \u0442\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u044d\u0442\u043e \u00ab\u0443\u0434\u043e\u0431\u043d\u043e\u00bb \u0434\u043b\u044f \u043f\u043e\u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0445 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043e\u043a.<\/p>\n<h3>\u0420\u0435\u0448\u0435\u043d\u0438\u0435 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438<\/h3>\n<p style=\"text-align:justify;\">\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0432 \u0444\u0443\u043d\u043a\u0446\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u0432 \u044d\u0442\u043e \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/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;\">\u0427\u0442\u043e\u0431\u044b \u0440\u0435\u0448\u0438\u0442\u044c \u044d\u0442\u043e\u0442 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b, \u043c\u043e\u0436\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0435 \u043f\u043e\u0434\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0443<\/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;\">\u0421 \u044d\u0442\u0438\u043c \u0438\u043c\u0435\u0435\u043c:<\/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;\">\u041c\u043e\u0436\u043d\u043e \u0437\u0430\u043c\u0435\u0442\u0438\u0442\u044c, \u0447\u0442\u043e \u043a\u0440\u0438\u0432\u0430\u044f \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0430 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0430 \u043a\u0430\u043a \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u043a\u0440\u0438\u0432\u0430\u044f \u0432 \u043f\u043e\u043b\u044f\u0440\u043d\u044b\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430\u0445, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0441 \u0446\u0438\u043a\u043b\u043e\u0439\u0434\u043e\u0439, \u043d\u0430\u0447\u0438\u043d\u0430\u044e\u0449\u0435\u0439\u0441\u044f \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442.<\/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;\">\u041a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> \u0431\u044b\u043b\u0430 \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e, \u0447\u0442\u043e\u0431\u044b \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u0438\u0442\u044c \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0435, \u0447\u0442\u043e \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u044f \u043d\u0430\u0447\u0438\u043d\u0430\u0435\u0442\u0441\u044f \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u043c\u043e\u0436\u043d\u043e \u0437\u0430\u043c\u0435\u0442\u0438\u0442\u044c, \u0447\u0442\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043f\u0430\u0440\u0430 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0442 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u044b\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438, \u0433\u0434\u0435 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u043c\u043e\u0436\u043d\u043e \u043d\u0430\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0442\u0430\u043a, \u0447\u0442\u043e\u0431\u044b \u043a\u0440\u0438\u0432\u0430\u044f \u043f\u0440\u043e\u0445\u043e\u0434\u0438\u043b\u0430 \u0447\u0435\u0440\u0435\u0437 \u0442\u043e\u0447\u043a\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)<\/span> \u0432 \u043a\u043e\u043d\u0446\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438. \u042d\u0442\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f:<\/p>\n<p style=\"text-align:center;\"><strong>\u0412\u0430\u0440\u0438\u0430\u043d\u0442 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>\u0412\u0430\u0440\u0438\u0430\u043d\u0442 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;\">\u0416\u0438\u0437\u043d\u0435\u0441\u043f\u043e\u0441\u043e\u0431\u043d\u043e\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u0435 \u0434\u043b\u044f \u044d\u0442\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438 \u0434\u0430\u0435\u0442\u0441\u044f \u0432\u0442\u043e\u0440\u044b\u043c \u0432\u0430\u0440\u0438\u0430\u043d\u0442\u043e\u043c, \u0438, \u043d\u0430\u0441\u0442\u0440\u043e\u0438\u0432 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u043a\u0430\u043a \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u043a\u0440\u0438\u0432\u0443\u044e, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u0435\u0442 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u044b\u043c \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f.<\/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=\"\u041f\u0440\u0438\u043c\u0435\u0440 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f, \u0434\u0443\u0433\u0430 \u0446\u0438\u043a\u043b\u043e\u0438\u0434\u0430\" 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=\"\u041f\u0440\u0438\u043c\u0435\u0440 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f, \u0434\u0443\u0433\u0430 \u0446\u0438\u043a\u043b\u043e\u0438\u0434\u0430\" width=\"497\" height=\"329\" class=\"aligncenter size-full wp-image-28731 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/solbraquis.png 497w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/solbraquis-300x199.png 300w\" sizes=\"(max-width: 497px) 100vw, 497px\" \/><\/noscript><\/center><\/p>\n<h3>\u041e\u043a\u043e\u043d\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043d\u0430\u0441\u0442\u0440\u043e\u0439\u043a\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f<\/h3>\n<p style=\"text-align:justify;\">\u041f\u043e\u0441\u043b\u0435 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0445 \u043d\u0430\u0441\u0442\u0440\u043e\u0435\u043a \u043a\u0440\u0438\u0432\u0430\u044f \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0430 \u0438\u043c\u0435\u0435\u0442 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u0444\u043e\u0440\u043c\u0443:<\/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;\">\u0417\u0434\u0435\u0441\u044c \u0431\u044b\u043b\u043e \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u043e <span class=\"katex-eq\" data-katex-display=\"false\">a=-b,<\/span> \u0433\u0434\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">0\\lt b.<\/span> \u041a\u0440\u0438\u0432\u0430\u044f \u0438\u043c\u0435\u0435\u0442 \u043f\u0435\u0440\u0438\u043e\u0434 <span class=\"katex-eq\" data-katex-display=\"false\">2b\\pi<\/span> \u0438 \u0434\u043e\u043b\u0436\u043d\u0430 \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u0442\u044c \u0443\u0441\u043b\u043e\u0432\u0438\u044e <span class=\"katex-eq\" data-katex-display=\"false\">x_2 \\in ]0,2b\\pi[<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">y_2 \\in ]-2b,0[.<\/span> \u041f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u0435 \u043e\u0441\u043e\u0431\u0435\u043d\u043d\u043e \u0432\u0430\u0436\u043d\u043e, \u0442\u0430\u043a \u043a\u0430\u043a \u0442\u0440\u0435\u0431\u0443\u0435\u0442, \u0447\u0442\u043e\u0431\u044b \u043a\u0440\u0438\u0432\u0430\u044f \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u043b\u0430\u0441\u044c \u043a\u0430\u043a \u043e\u0434\u043d\u0430 \u0434\u0443\u0433\u0430 \u0446\u0438\u043a\u043b\u043e\u0438\u0434\u0430, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0440\u0435\u0448\u0435\u043d\u0438\u0435 \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043d\u0435\u0442 \u0431\u044b\u0442\u044c \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u043c, \u0435\u0441\u043b\u0438 \u0447\u0430\u0441\u0442\u0438\u0446\u0430 \u0441\u043d\u043e\u0432\u0430 \u0432\u0435\u0440\u043d\u0435\u0442\u0441\u044f \u0432 \u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043f\u043e\u043a\u043e\u044f \u043f\u0440\u0438 \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0435\u043d\u0438\u0438 \u043a \u0442\u043e\u0447\u043a\u0435 \u043d\u0443\u043b\u0435\u0432\u043e\u0439 \u0432\u044b\u0441\u043e\u0442\u044b.<\/p>\n<p style=\"text-align:justify;\">\u0427\u0442\u043e\u0431\u044b \u043d\u0430\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u044d\u0442\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u0437\u0430\u0434\u0430\u0447\u0438, \u043d\u0430\u043c \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u043d\u0430\u0439\u0442\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u044e\u0442 \u0441\u0438\u0441\u0442\u0435\u043c\u0435:<\/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;\">\u042d\u0442\u0430 \u043d\u0435\u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u0430, \u043f\u043e\u0445\u043e\u0436\u0435, \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0440\u0435\u0448\u0435\u043d\u0438\u0439, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u0447\u0438\u0441\u043b\u0435\u043d\u043d\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b \u0432 Wolfram Mathematica. \u0414\u0430\u043b\u0435\u0435 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u044b \u0448\u0430\u0433\u0438 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438:<\/p>\n<p><\/br><\/p>\n<h4>\u0428\u0430\u0433 1: \u0423\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u044c \u0441\u0438\u0441\u0442\u0435\u043c\u0443<\/h4>\n<p style=\"text-align:justify;\">\u0423\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u044c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u0441\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0449\u0438\u0435 \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f<\/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>\u0428\u0430\u0433 2: \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c \u043a\u043e\u043d\u0435\u0447\u043d\u0443\u044e \u0442\u043e\u0447\u043a\u0443<\/h4>\n<p style=\"text-align:justify;\">\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c \u0442\u043e\u0447\u043a\u0443, \u0432 \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u0447\u0430\u0441\u0442\u0438\u0446\u0430 \u043f\u0440\u0438\u0431\u0443\u0434\u0435\u0442 \u0432 \u043a\u043e\u043d\u0446\u0435 \u0441\u0432\u043e\u0435\u0433\u043e \u043f\u0443\u0442\u0438. \u0412 \u0434\u0430\u043d\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u043c \u0435\u0435 \u0432 <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)=(1,-2).<\/span> \u042d\u0442\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u043c\u043e\u0436\u043d\u043e \u0438\u0437\u043c\u0435\u043d\u0438\u0442\u044c, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043f\u0440\u043e\u0431\u043e\u0432\u0430\u0442\u044c \u0434\u0440\u0443\u0433\u0438\u0435 \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u044b\u0435 \u043a\u043e\u043d\u0444\u0438\u0433\u0443\u0440\u0430\u0446\u0438\u0438.<\/p>\n<p><code>x2val = 1; y2val = -2;<\/code><br \/>\n<\/br><\/p>\n<h4>\u0428\u0430\u0433 3: \u0427\u0438\u0441\u043b\u0435\u043d\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0438\u0441\u043a\u043e\u043c\u044b\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f<\/h4>\n<p style=\"text-align:justify;\">\u0418\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u00abFindRoot\u00bb \u0434\u043b\u044f \u0447\u0438\u0441\u043b\u0435\u043d\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438<\/p>\n<p><code>sol = FindRoot[{eq1, eq2} \/. {x2 -> x2val, y2 -> y2val}, {{b,1}, {theta, 1}}]<\/code><\/p>\n<p style=\"text-align:justify;\">\u0417\u0434\u0435\u0441\u044c \u0431\u044b\u043b\u0438 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u044b \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">b=1<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta=1<\/span> \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0438 \u0434\u043b\u044f \u0447\u0438\u0441\u043b\u0435\u043d\u043d\u043e\u0433\u043e \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f. \u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0440\u0435\u0448\u0435\u043d\u0438\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">b\\approx 2.4056<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta \\approx 1.40138<\/span>\n<p><\/br><\/p>\n<h4>\u0428\u0430\u0433 4: \u041f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432<\/h4>\n<p style=\"text-align:justify;\">\u041d\u0430\u043f\u043e\u043c\u043d\u0438\u043c, \u0447\u0442\u043e \u0434\u043b\u044f \u0442\u043e\u0433\u043e \u0447\u0442\u043e\u0431\u044b \u044d\u0442\u0438 \u043e\u0442\u0432\u0435\u0442\u044b \u0438\u043c\u0435\u043b\u0438 \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0441\u043c\u044b\u0441\u043b, \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e, \u0447\u0442\u043e\u0431\u044b <span class=\"katex-eq\" data-katex-display=\"false\">x_2 \\in ]0,2b\\pi[<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">y_2 \\in ]-2b, 0[.<\/span> \u041c\u044b \u043c\u043e\u0436\u0435\u043c \u0431\u044b\u0441\u0442\u0440\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c, \u0447\u0442\u043e \u044d\u0442\u043e \u0442\u0430\u043a, \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0433\u043e \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u0430<\/p>\n<p style=\"text-align:justify;\">\u0421\u043d\u0430\u0447\u0430\u043b\u0430 \u0438\u0437\u0432\u043b\u0435\u043a\u0430\u0435\u043c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span>, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u044f<\/p>\n<p><code>bval = sol[[1, 2]]; thetaval = sol[[2, 2]];<\/code><\/p>\n<p style=\"text-align:justify;\">\u0417\u0430\u0442\u0435\u043c \u0434\u0430\u0435\u043c \u043a\u043e\u043c\u0430\u043d\u0434\u0443 \u043d\u0430 \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u043f\u043e\u0434\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u044f<\/p>\n<p><code>If[0 < x2val < 2*Pi*bval &#038;&#038; -2*bval < y2val < 0 \"\u0414\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\", \"\u041d\u0435\u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\"]\n<\/code><\/p>\n<p style=\"text-align:justify;\">\u0415\u0441\u043b\u0438 \u0432\u0441\u0435 \u043f\u0440\u043e\u0448\u043b\u043e \u0445\u043e\u0440\u043e\u0448\u043e, \u043c\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \"\u0414\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\" \u0432 \u0432\u044b\u0432\u043e\u0434\u0435. \u042d\u0442\u043e\u0442 \u0444\u0440\u0430\u0433\u043c\u0435\u043d\u0442 \u043a\u043e\u0434\u0430 \u043f\u043e\u043c\u043e\u0436\u0435\u0442 \u0432\u0430\u043c \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c, \u043f\u0440\u0430\u0432\u0438\u043b\u044c\u043d\u043e \u043b\u0438 \u0441\u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u0430 \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044f.<\/p>\n<p style=\"text-align:justify;\">\u0421 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u044d\u0442\u0438\u0445 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440 \u0443 \u043d\u0430\u0441 \u043d\u0430\u043a\u043e\u043d\u0435\u0446-\u0442\u043e \u043f\u043e\u043b\u043d\u043e\u0441\u0442\u044c\u044e \u043d\u0430\u0441\u0442\u0440\u043e\u0435\u043d\u0430 \u043d\u0430\u0448\u0430 \u043a\u0440\u0438\u0432\u0430\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0441\u043e\u0435\u0434\u0438\u043d\u044f\u0435\u0442 \u0442\u043e\u0447\u043a\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">(x_1,y_1)=(0,0)<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">(x_2,y_2)=(1,-2).<\/span> \u041f\u043e\u043b\u0443\u0447\u0438\u0432\u0448\u0430\u044f\u0441\u044f \u043a\u0440\u0438\u0432\u0430\u044f:<\/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;\">\u0413\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u0438 \u044d\u0442\u043e \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0442\u0430\u043a:<\/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=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/sol2braquis.png 296w, https:\/\/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>\u0420\u0435\u043f\u043e\u0437\u0438\u0442\u043e\u0440\u0438\u0439 GitHub \u0441 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u043c Wolfram<\/h2>\n<p style=\"text-align:justify;\">\u041f\u043e\u043b\u043d\u044b\u0439 \u043a\u043e\u0434 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435, \u0432\u043a\u043b\u044e\u0447\u0430\u044f \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c, \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0430\u043d\u043d\u044b\u0439 \u0432 Wolfram Mathematica, \u0434\u043e\u0441\u0442\u0443\u043f\u0435\u043d \u0434\u043b\u044f \u0441\u043a\u0430\u0447\u0438\u0432\u0430\u043d\u0438\u044f \u0438 \u043e\u0437\u043d\u0430\u043a\u043e\u043c\u043b\u0435\u043d\u0438\u044f \u0432 \u043c\u043e\u0435\u043c \u0440\u0435\u043f\u043e\u0437\u0438\u0442\u043e\u0440\u0438\u0438 GitHub. \u042d\u0442\u043e\u0442 \u0440\u0435\u043f\u043e\u0437\u0438\u0442\u043e\u0440\u0438\u0439 \u0432\u043a\u043b\u044e\u0447\u0430\u0435\u0442 \u0432 \u0441\u0435\u0431\u044f \u0444\u0430\u0439\u043b `.nb` \u0441 \u043a\u043e\u0434\u043e\u043c \u0432 \u0444\u043e\u0440\u043c\u0430\u0442\u0435 \u0438\u043d\u0442\u0435\u0440\u0430\u043a\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u0431\u043b\u043e\u043a\u043d\u043e\u0442\u0430, \u0430 \u0442\u0430\u043a\u0436\u0435 \u0432\u0435\u0440\u0441\u0438\u044e \u0432 \u0432\u0438\u0434\u0435 \u043f\u0440\u043e\u0441\u0442\u043e\u0433\u043e \u0442\u0435\u043a\u0441\u0442\u0430 `.m` \u0434\u043b\u044f \u0442\u0435\u0445, \u043a\u0442\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u0447\u0438\u0442\u0430\u0435\u0442 \u043f\u0440\u043e\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044c \u043a\u043e\u0434 \u043d\u0430\u043f\u0440\u044f\u043c\u0443\u044e.<\/p>\n<p style=\"text-align:justify;\"><strong>\u0412\u044b \u043c\u043e\u0436\u0435\u0442\u0435 \u0441\u043a\u0430\u0447\u0430\u0442\u044c \u0440\u0435\u043f\u043e\u0437\u0438\u0442\u043e\u0440\u0438\u0439 \u0441 GitHub <a href=\"https:\/\/github.com\/girebz\/Braquist-crona\" target=\"_blank\" rel=\"noopener\">\u0437\u0434\u0435\u0441\u044c<\/a>.<\/strong><\/p>\n<p style=\"text-align:justify;\">\u041f\u043e\u043c\u0438\u043c\u043e \u043a\u043e\u0434\u0430, \u0440\u0435\u043f\u043e\u0437\u0438\u0442\u043e\u0440\u0438\u0439 \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u0442 \u0444\u0430\u0439\u043b \"README\" \u0441 \u043f\u043e\u0434\u0440\u043e\u0431\u043d\u044b\u043c\u0438 \u0438\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f\u043c\u0438 \u043f\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u044e \u0438 \u043f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u044e \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430, \u0430 \u0442\u0430\u043a\u0436\u0435 \u043f\u043e\u0448\u0430\u0433\u043e\u0432\u043e\u0435 \u043e\u0431\u044a\u044f\u0441\u043d\u0435\u043d\u0438\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438 \u043e \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u0435. \u041d\u0430\u0434\u0435\u044e\u0441\u044c, \u044d\u0442\u043e \u0431\u0443\u0434\u0435\u0442 \u043f\u043e\u043b\u0435\u0437\u043d\u043e!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0412\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0435 \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u0432 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0435 \u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 \u0420\u0435\u0437\u044e\u043c\u0435:\u0412 \u044d\u0442\u043e\u043c \u0443\u0440\u043e\u043a\u0435 \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0432\u044b\u0432\u043e\u0434 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 \u0438\u0437 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0438 \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0433\u043e \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f, \u0438 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u044d\u0442\u043e\u0433\u043e \u043f\u043e\u0434\u0440\u043e\u0431\u043d\u043e \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0435\u0433\u043e \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0434\u043b\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u044b \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u044b. \u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f: \u041f\u043e \u0437\u0430\u0432\u0435\u0440\u0448\u0435\u043d\u0438\u0438 \u044d\u0442\u043e\u0433\u043e \u0443\u0440\u043e\u043a\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442: \u041f\u043e\u043d\u044f\u0442\u044c \u043f\u0440\u0438\u043d\u0446\u0438\u043f \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0435\u0433\u043e \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430 \u041f\u0440\u043e\u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 \u0420\u0435\u0448\u0438\u0442\u044c \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0443 \u0431\u0440\u0430\u0445\u0438\u0441\u0442\u043e\u0445\u0440\u043e\u043d\u044b, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28745,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":55,"footnotes":""},"categories":[946,645],"tags":[],"class_list":["post-28789","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-946","category-645"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ 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