{"id":2231,"date":"2017-03-11T17:11:30","date_gmt":"2017-03-11T09:11:30","guid":{"rendered":"http:\/\/www.systemsci.org\/jinshanw\/?p=2231"},"modified":"2017-03-15T10:07:27","modified_gmt":"2017-03-15T02:07:27","slug":"%e5%b8%ae%e5%a5%bd%e5%a5%87%e5%ae%9d%e5%ae%9d%e5%90%8c%e5%ad%a6%e8%a7%a3%e7%ad%94%e4%b8%80%e4%b8%aa%e7%89%a9%e7%90%86%e9%a2%98","status":"publish","type":"post","link":"https:\/\/www.systemsci.org\/jinshanw\/2017\/03\/11\/%e5%b8%ae%e5%a5%bd%e5%a5%87%e5%ae%9d%e5%ae%9d%e5%90%8c%e5%ad%a6%e8%a7%a3%e7%ad%94%e4%b8%80%e4%b8%aa%e7%89%a9%e7%90%86%e9%a2%98\/","title":{"rendered":"\u5e2e\u597d\u5947\u5b9d\u5b9d\u540c\u5b66\u89e3\u7b54\u4e00\u4e2a\u7269\u7406\u9898"},"content":{"rendered":"

\u6628\u5929\u6536\u5230\u540c\u5b66\u53d1\u8fc7\u6765\u7684\u4e00\u9053\u9898\uff1a
\n\"-storage-emulated-0-tencent-MicroMsg-31fda0f8bcdbd04c7987df360e6c4f06-image2-05-66-056677ebeacee6170468ddec13fa2650.temp\"
\n\u4ee5\u53ca\u6807\u51c6\u7b54\u6848\uff1a
\n\"-storage-emulated-0-tencent-MicroMsg-31fda0f8bcdbd04c7987df360e6c4f06-image2-a1-b6-a1b660f97cd23815529eccc5cc13d9c5\"<\/p>\n

\u5b9e\u9645\u7269\u7406\u8fc7\u7a0b\u6bd4\u7b54\u6848\u590d\u6742\u4e00\u70b9\uff0c\u9700\u8981\u8003\u8651\u7535\u573a\u529b\u548c\u7a7a\u6c14\u963b\u529b\u7684\u5408\u529b\uff0c\u4e5f\u5c31\u662f
\n\[
\nm\\ddot{x} + k\\dot{x}-\\frac{URq}{H}\\frac{1}{x}=0.
\n\]
\n\u8fd9\u4e2a\u65b9\u7a0b\u5982\u679c\u771f\u60f3\u6c42\u89e3\uff0c\u5f88\u56f0\u96be\u7684\u3002\u56e0\u6b64\uff0c\u53ef\u80fd\u6709\u7b80\u5355\u4e00\u70b9\u7684\u529e\u6cd5\u3002\u8003\u8651\u7269\u7406\u56fe\u666f\uff1a\u5c18\u57c3\u4e00\u5f00\u59cb\u901f\u5ea6\u4e3a\u96f6\uff0c\u6ca1\u6709\u963b\u529b\uff0c\u63a5\u7740\u6536\u5230\u7535\u573a\u529b\uff0c\u53ef\u80fd\u52a0\u901f\u5ea6\u6bd4\u8f83\u5927\uff0c\u4e8e\u662f\u5f88\u77ed\u7684\u65f6\u95f4\u5185\u5c31\u6709\u53ef\u80fd\u88ab\u52a0\u901f\uff0c\u4e00\u76f4\u5230\u7a7a\u6c14\u963b\u529b\u4e0d\u53ef\u5ffd\u7565\uff0c\u6700\u7ec8\u4f1a\u5230\u8fbe\u4e00\u4e2a\u6781\u503c\u3002\u8fd9\u4e2a\u6781\u503c\u6ee1\u8db3
\n\[
\n\\dot{x}-\\frac{URq}{Hk}\\frac{1}{x}=0 \\Longrightarrow v_{max}\\left(x\\right) = \\frac{URq}{Hk}\\frac{1}{x}.
\n\]
\n\u63a5\u7740\u6211\u4eec\u505a\u4e00\u4e2a\u7b80\u5355\u63a8\u5e7f\uff0c\u5c31\u53ef\u4ee5\u731c\u60f3\u5982\u679c\u8fd9\u4e2a\u901f\u5ea6\u548c\u4f4d\u7f6e\u7684\u5173\u7cfb\u4e00\u76f4\u90fd\u5bf9\uff0c\u90a3\u4e48\uff0c\u4e0a\u9762\u7684\u7269\u7406\u9898\u5c31\u641e\u5b9a\u4e86\u3002\u4f46\u662f\uff0c\u8003\u8651\u5c18\u57c3\u7b2c\u4e00\u6b21\u5230\u8fbe\u8fd9\u4e2a\u901f\u5ea6\u4e4b\u540e\uff08\u8fd9\u4e2a\u95ee\u9898\u4e0d\u5927\uff0c\u53ef\u4ee5\u5047\u8bbe\u65f6\u95f4\u5f88\u77ed\u8def\u7a0b\u5f88\u77ed\uff0c\u8fd9\u4e9b\u5047\u8bbe\u8fd8\u53ef\u4ee5\u68c0\u9a8c\uff09\uff0c\u7a7a\u6c14\u963b\u529b\u5c31\u5df2\u7ecf\u5f88\u5927\u4e86\uff0c\u8fd9\u4e2a\u65f6\u5019\u5e94\u8be5\u6709\u4e00\u6bb5\u65f6\u95f4\u901f\u5ea6\u4f1a\u51cf\u5c11\uff0c\u800c\u4e14\u4e0d\u4e00\u5b9a\u5c31\u80fd\u5b9e\u73b0\u4e0a\u9762\u8fd9\u4e2a\u201c\u51c6\u5e73\u8861\u201d\u6761\u4ef6\u3002\u5f53\u7136\uff0c\u5230\u5e95\u662f\u5426\u80fd\u591f\u5b9e\u73b0\uff0c\u5c31\u8981\u628a\u7cbe\u786e\u89e3\u6c42\u51fa\u6765\u548c\u8fd9\u4e2a\u731c\u60f3\u6bd4\u8f83\u4e00\u4e0b\u3002\u6211\u672c\u6765\u63d0\u4e00\u63d0\u8fd9\u4e2a\u8fd8\u9700\u8981\u8fdb\u4e00\u6b65\u9a8c\u8bc1\u5c31\u591f\u4e86\u3002\u4f46\u662f\uff0c\u6211\u90a3\u4e2a\u540c\u5b66\u662f\u4e2a\u597d\u5947\u5b9d\u5b9d\uff0c\u975e\u5f97\u8ba9\u6211\u7b97\u51fa\u6765\u3002\u597d\u5427\uff0c\u4e0b\u9762\u662f\u8ba1\u7b97\u7ed3\u679c\u3002<\/p>\n

\u9996\u5148\uff0c\u7ed9\u65b9\u7a0b\u505a\u4e2a\u91cd\u6807\u5ea6\uff0c\(x\\rightarrow \\frac{x}{R}\)\uff0c\u53bb\u6389\u5355\u4f4d\u53d8\u6210\(\\left[0,1\\right]\)\u4e4b\u95f4\u7684\u7eaf\u6570\uff08\u5176\u5b9e\u8fd8\u6709\u66f4\u597d\u7684\u65e0\u91cf\u7eb2\u7ec4\u5408\uff0c\u6682\u65f6\u5c31\u5148\u7528\u8fd9\u4e2a\u4e86\uff09\uff0c\u5e76\u4e14\u5206\u5f00\u6210\u4e3a\u4e00\u9636\u65b9\u7a0b\uff0c\u5f97\u5230
\n\begin{align}
\n\\dot{x} = y, \
\n\\dot{y} = -a y+ \\frac{b}{x}.
\n\end{align}
\n\u5176\u4e2d\(a=\\frac{k}{m}, b=\\frac{Uq}{HRm}=\\frac{Uq}{HRk}a\)\u3002\u6ce8\u610f\uff0c\u7531\u4e8e\u6ca1\u6709\u5b8c\u5168\u628a\u65b9\u7a0b\u65e0\u91cf\u7eb2\u5316\uff08\u90a3\u4e2a\u65f6\u5019\u65f6\u95f4\u7a7a\u95f4\u7684\u5355\u4f4d\u90fd\u8981\u53bb\u6389\uff0c\u73b0\u5728\u5c31\u53bb\u6389\u4e86\u7a7a\u95f4\uff09\uff0c\u8fd9\u91cc\u7684a,b\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u5173\u7cfb\u662f\u9700\u8981\u505a\u8ba8\u8bba\u4e86\u3002\u4e0d\u8fc7\uff0c\u76f8\u5f53\u4e8e\u8981\u6362\u4e00\u4e0b\u6211\u4e0b\u9762\u7ed9\u51fa\u6765\u7684\u56fe\u91cc\u9762\u7684\u65f6\u95f4\u7684\u5355\u4f4d\u800c\u5df2\uff0c\u5c31\u5077\u61d2\u4e86\u3002<\/p>\n

\u653e\u5230SageMath\u91cc\u9762\uff0c\u7528Runge-Kutta4\u65b9\u6cd5\u6c42\u89e3\u5f97\u5230\u4e0b\u9762\u7684\u56fe\u3002<\/p>\n

\n from sage.calculus.desolvers import desolve_system_rk4
\n x,y,t=var(‘x y t’)
\n a=1.0
\n b=0.01
\n P=desolve_system_rk4([y,-a*y+b\/x],[x,y],ics=[0,0.01,0],ivar=t,end_points=20.0,step=0.001)
\n \u5982\u679c\u4f60\u60f3\u4eb2\u81ea\u6765\u8fd0\u884c\u8fd9\u6bb5\u7a0b\u5e8f\uff0c\u70b9\u8fd9\u91cc<\/a>\n<\/p><\/blockquote>\n

\u4f4d\u79fb\u65f6\u95f4\u66f2\u7ebf\uff1a
\n\"xt\"<\/p>\n

\u901f\u5ea6\u65f6\u95f4\u66f2\u7ebf\uff1a
\n\"vt\"<\/p>\n

\u901f\u5ea6\u4f4d\u79fb\u66f2\u7ebf\uff0c\u731c\u60f3\u7684\u89e3\u548c\u6570\u503c\u89e3\u7684\u5bf9\u6bd4\uff1a
\n\"comparison\"<\/p>\n

\u53ef\u4ee5\u770b\u5230\uff0c\u786e\u5b9e\u4e0a\u5347\u548c\u4e0b\u964d\u671f\u95f4\u76f8\u5dee\u6bd4\u8f83\u5927\uff0c\u4f46\u662f\uff0c\u5728\u540e\u671f\u6bd4\u8f83\u63a5\u8fd1\u3002\u6211\u8fd8\u5c1d\u8bd5\u4e86\u5176\u4ed6\u7684\u53c2\u6570\u60c5\u51b5\uff0c\u4f8b\u5982\(a=1.0, b=1.0\)\uff0c\(a=1.0, b=10.0\)\uff0c\u90fd\u5f97\u5230\u4e00\u6837\u7684\u957f\u65f6\u72b6\u6001\uff0c\u6570\u503c\u89e3\u548c\u731c\u6d4b\u89e3\u5728\u540e\u9762\u76f8\u7b26\u3002
\n\"a1b10\"<\/p>\n

\u7ed3\u8bba\uff1a\u53ea\u8981\u5141\u8bb8\u5c18\u57c3\u8fd0\u52a8\u65f6\u95f4\u8db3\u591f\u957f\uff0c\u4e5f\u5c31\u662fR\u8db3\u591f\u5927\uff0c\u90a3\u4e48\uff0c\u731c\u6d4b\u89e3\u8fd8\u662f\u5408\u7406\u7684\uff0c\u5e76\u4e14\u8fd9\u4e2a\u7ed3\u679c\u5bf9\u4e8e\u53c2\u6570\u7684\u60c5\u51b5\u5f88\u9c81\u68d2\u3002\u8fd9\u4e2a\u548c\u6211\u901a\u8fc7\u7269\u7406\u56fe\u666f\u4f30\u8ba1\u7684\u4e0d\u592a\u7b26\u5408\u3002\u6211\u4ee5\u4e3a\uff0c\u6709\u4e00\u4e9b\u53c2\u6570\u60c5\u51b5\u4f1a\u4e24\u8005\u76f8\u7b26\uff0c\u6709\u4e00\u4e9b\u60c5\u51b5\u4f1a\u4e24\u8005\u76f8\u5dee\u5f88\u5927\u3002\u5f53\u7136\uff0c\u6211\u4eec\u770b\u5230\u6709\u7684\u65f6\u5019\u5728\u77ed\u65f6\u95f4\u5185\u76f8\u5dee\u4e5f\u5f88\u5927\uff0c\u4f46\u662f\uff0c\u957f\u671f\u6765\u770b\uff0c\u603b\u662f\u5411\u7740\u731c\u6d4b\u89e3\u7684\u65b9\u5411\u5728\u903c\u8fd1\u3002<\/p>\n

\u8ba8\u8bba\uff1a\u8fd9\u4e2a\u73b0\u8c61\u5176\u5b9e\u5f88\u6709\u610f\u601d\u3002\u5f53\u6211\u5f97\u5230\u8fd9\u4e2a\u7ed3\u8bba\u4e4b\u540e\uff0c\u6211\u60f3\u4e86\u534a\u5929\u3002\u5b9e\u9645\u4e0a\uff0c\u7a7a\u6c14\u963b\u529b\u548c\u7535\u573a\u529b\u76f8\u5f53\u4e8e\u4e24\u4e2a\u76f8\u4e92\u7ade\u4e89\u7684\u529b\uff0c\u800c\u4e14\u8fd9\u4e24\u4e2a\u529b\u7684\u7ade\u4e89\u4e0d\u662f\u4e00\u4e2a\u76d6\u8fc7\u53e6\u4e00\u4e2a\u4e4b\u540e\u5c31\u5b8c\u5168\u538b\u5236\u4e86\u88ab\u76d6\u8fc7\u7684\u90a3\u4e2a\uff0c\u800c\u662f\u5f3a\u7684\u4e00\u65b9\u4f1a\u8870\u51cf\u5f31\u7684\u4e00\u65b9\u4f1a\u6210\u957f\uff0c\u4e5f\u5c31\u662f\u8d1f\u53cd\u9988\u3002\u56e0\u6b64\uff0c\u53ea\u8981\u662f\u6709\u51c6\u5e73\u8861\u5b58\u5728\u7684\u8d1f\u53cd\u9988\uff0c\u7cfb\u7edf\u603b\u662f\u4f1a\u51b2\u7740\u90a3\u4e2a\u51c6\u5e73\u8861\u53bb\u6f14\u5316\u7684\u3002\u7ec8\u4e8e\u60f3\u660e\u767d\u4e86\u3002\u5ffd\u7136\u624d\u53d1\u73b0\uff0c\u5b66\u4e86\u591a\u5e74\u7684\u8d1f\u53cd\u9988\u548c\u5e73\u8861\u6001\uff0c\u8fd8\u662f\u6ca1\u6709\u771f\u7684\u8fdb\u5165\u601d\u8003\u6a21\u5f0f\uff0c\u5982\u679c\u8fdb\u5165\u4e86\uff0c\u5c31\u5e94\u8be5\u9884\u671f\u6709\u8fd9\u4e2a\u957f\u65f6\u72b6\u6001\u3002\u6211\u7684\u53e6\u4e00\u4e2a\u597d\u5947\u5b9d\u5b9d\u540c\u5b66\u5c31\u9884\u671f\u5230\u80af\u5b9a\u6709\u8fd9\u4e2a\u957f\u65f6\u72b6\u6001\uff0c\u5c3d\u7ba1\u6211\u4e0d\u6e05\u695a\u4ed6\u7684\u9884\u671f\u7684\u57fa\u7840\u3002\u5f53\u7136\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u5f53\u4e2d\uff08\u8fd9\u4e2a\u8003\u9898\u6709\u5b9e\u9645\u80cc\u666f\uff09\u5e94\u8be5\u662f\u5148\u6709\u4e00\u4e2a\u5bf9\u957f\u65f6\u72b6\u6001\u7684\u4f30\u8ba1\uff0c\u7136\u540e\uff0c\u518d\u7528\u6570\u503c\u8ba1\u7b97\u6765\u786e\u8ba4\uff0c\u7136\u540e\u5728\u8fdb\u5165\u5de5\u7a0b\u8bbe\u8ba1\u548c\u5b9e\u65bd\u3002<\/p>\n

\u603b\u7ed3\uff1a\u8d1f\u53cd\u9988\u7684\u6982\u5ff5\u5f88\u91cd\u8981\u3002\u6570\u503c\u8ba1\u7b97\u548c\u601d\u8003\u76f8\u4e92\u914d\u5408\u80fd\u591f\u628a\u95ee\u9898\u60f3\u5f97\u66f4\u6e05\u695a\u3002\u987a\u4fbf\u63a8\u8350\u4e00\u4e0bSageMath<\/a>\u8fd9\u4e2a\u6570\u5b66\u8f6f\u4ef6\uff0c\u5f88\u597d\u7528\uff0c\u5b8c\u5168\u514d\u8d39\uff0c\u8fd8\u514d\u5b89\u88c5\u76f4\u63a5\u5728\u7f51\u7edc\u4e0a\u8fd0\u884c\u3002\u4e0d\u8fc7\uff0c\u51fa\u9898\u4eba\u8fd8\u662f\u5e94\u8be5\u63d0\u793a\u4e00\u4e0b\uff0c\u4f8b\u5982\uff1a\u8003\u8651\u5230R\u5f88\u5927\u5f88\u5927\uff0c\u53ef\u4ee5\u4ec5\u8003\u8651\u67d0\u79cd\u5e73\u8861\u6001\u3002<\/p>\n

\u52a0\u4e00\u4e2a\u6ce8\uff1a\u5982\u679c\u7535\u8377\u5e26\u4e0a\u8d1f\u7535\u8377\uff0c\u5411\u7740\u4e2d\u5fc3\u8d70\uff0c\u8fd8\u662f\u6709\u4e24\u4e2a\u56e0\u7d20\u7ade\u4e89\uff0c\u4f46\u662f\u5176\u4e2d\u4e00\u4e2a\u6210\u4e86\u6b63\u53cd\u9988(\u7535\u573a\u529b\u7684\u6548\u679c\uff0c\u8d8a\u5f80\u91cc\u901f\u5ea6\u8d8a\u5927\u53d7\u529b\u4e5f\u8d8a\u5927\uff0c\u53ea\u80fd\u9760\u963b\u529b\u6765\u5e73\u8861\u4e00\u4e0b\uff0c\u4f46\u662f\u4e5f\u53ef\u80fd\u4f1a\u7ee7\u7eed\u53d8\u5927)\uff0c\u60c5\u51b5\u5c31\u4f1a\u5b8c\u5168\u4e0d\u4e00\u6837\u4e86\u3002\u89c1\u4e0a\u9762\u90a3\u4e2aSageMath\u7a0b\u5e8f<\/a>\u7684\u540e\u534a\u90e8\u5206\uff0c\u8d1f\u7535\u8377\u3002<\/p>\n

\u52a0\u7b2c\u4e8c\u4e2a\u6ce8\uff1a\u505a\u4e86\u4e00\u4e0b\u5348\u7684\u9898\uff0c\u6536\u83b7\u4e86\u5bf9\u8d1f\u53cd\u9988\u7684\u66f4\u52a0\u6df1\u523b\u7684\u7406\u89e3\uff0c\u8ba9\u5b83\u66f4\u8fdb\u4e00\u6b65\u5730\u6210\u4e86\u601d\u8003\u7684\u57fa\u7840\u3002\u6211\u672c\u6765\u5c31\u662f\u6559\u7cfb\u7edf\u79d1\u5b66\u7684\uff0c\u7a33\u5b9a\u6027\u5206\u6790\u4e5f\u5728\u6211\u81ea\u5df1\u7684\u7814\u7a76\u5de5\u4f5c\u4e2d\u7528\u8fc7\u3002\u53ef\u662f\uff0c\u90a3\u4e2a\u7406\u89e3\u8fd8\u662f\u80a4\u6d45\u7684\uff0c\u548c\u6559\u79d1\u4e66\u4e0a\u7684\u592a\u50cf\u4e86\u3002\u76f4\u5230\u7b97\u5b8c\u8fd9\u4e2a\u9898\uff0c\u624d\u7406\u89e3\u66f4\u6df1\u523b\u4e86\u3002\u8fd9\u4e2a\u5e94\u8be5\u5206\u4eab\u7ed9\u5b66\u751f\uff0c\u7528\u6765\u8bf4\u660e\uff0c\u600e\u4e48\u505a\u4f5c\u4e1a\u4e3a\u4ec0\u4e48\u8981\u505a\u4f5c\u4e1a\uff0c\u505a\u5b8c\u4f5c\u4e1a\u4ee5\u540e\u7684\u601d\u8003\u6709\u591a\u91cd\u8981\u3002\u5982\u679c\u4e0d\u505a\u4f5c\u4e1a\uff0c\u800c\u4e14\u8fd8\u662f\u6709\u96be\u5ea6\u6709\u6df1\u5ea6\u7684\u4f5c\u4e1a\uff0c\u90a3\u4e48\u4ec5\u4ec5\u662f\u770b\u8d77\u6765\u5b66\u61c2\u4e86\uff0c\u4e0d\u662f\u771f\u7684\u61c2\u3002\u53ea\u6709\u8fdb\u4e00\u6b65\u7528\u4e86\u8fd9\u4e2a\uff0c\u800c\u4e14\u7528\u5230\u65b0\u7684\u5730\u65b9\u548c\u6559\u6750\u4e2d\u4f8b\u5b50\u4e0d\u4e00\u6837\u7684\u5730\u65b9\uff0c\u624d\u662f\u771f\u7684\u61c2\u3002<\/p>\n

\u52a0\u7b2c\u4e09\u4e2a\u6ce8\uff1a\u521a\u624d\u771f\u7684\u505a\u4e86\u4e00\u4e0b\u8fd9\u4e2a\u65b9\u7a0b\u7684\u7a33\u5b9a\u6027\u5206\u6790\uff0c\u800c\u4e14\u662f\u8f68\u9053\u7684\u7a33\u5b9a\u6027\u5206\u6790\uff0c\u4e0d\u662f\u4e0d\u52a8\u70b9\u7684\u7a33\u5b9a\u6027\u5206\u6790\u3002\u592a\u597d\u4e86\uff0c\u4ee5\u540e\u7814\u7a76\u751f\u671f\u672b\u8003\u8bd5\u8bd5\u9898\u53c8\u591a\u4e86\u4e00\u4e2a\u597d\u4f8b\u5b50\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\uff0c\u5efa\u8bae\u8bd5\u8bd5\u8fd9\u4e2a\u7a33\u5b9a\u6027\u5206\u6790\u3002\u901a\u8fc7\u7a33\u5b9a\u6027\u5206\u6790\u5f88\u5bb9\u6613\u5c31\u80fd\u591f\u770b\u5230\uff0c\u53ea\u8981b\u90e8\u5206\u7684\u7b26\u53f7\u53d8\u4e86\uff0c\u7cfb\u7edf\u7684\u7a33\u5b9a\u6027\u5c31\u53d1\u751f\u4e86\u5b9a\u6027\u7684\u53d8\u5316\u3002\u7b54\u6848\u6211\u5c31\u4e0d\u63ed\u6653\u5728\u8fd9\u91cc\u4e86\u3002<\/p>\n

\u8865\u5145\uff1a\u6709\u4eba\u95ee\uff0c\u90a3\u4e2a\u91cf\u7eb2\u5206\u6790\u505a\u5b8c\u4e86\u4f1a\u600e\u6837\uff0c\u80fd\u4e0d\u80fd\u5e2e\u6211\u4eec\u770b\u5f97\u66f4\u52a0\u6df1\u523b\u3002\u597d\u5427\uff0c\u6211\u6765\u505a\u5b8c\u5b83\u3002\u4e0d\u8ba9\u6211\u5077\u61d2\u7684\u8282\u594f\u3002<\/p>\n

\u8bb0\(\\gamma = \\frac{qUR}{H}\)\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e2d\u5404\u4e2a\u7269\u7406\u91cf\u7684\u91cf\u521a\u5982\u4e0b\uff0c
\n\\begin{align}
\n\\left[x\\right] = m, \
\n\\left[t\\right] = s, \
\n\\left[\\kappa\\right] = N\\frac{s}{m} = \\frac{kg}{s}, \
\n\\left[\\gamma\\right] = N\\cdot m = \\frac{kgm^{2}}{s^{2}}, \
\n\\left[m\\right] = kg.
\n\\end{align}
\n\u6211\u4eec\u5e0c\u671b\u505a\u4e00\u4e2a\u5355\u4f4d\u53d8\u6362\uff0c\u5c3d\u91cf\u628a\u53d8\u91cf\u53d8\u6210\u65e0\u91cf\u7eb2\u7684\u7eaf\u6570\u3002\u8fd9\u91cc\u53ea\u6709\u957f\u5ea6\u548c\u65f6\u95f4\u4e24\u4e2a\u5355\u4f4d\u9700\u8981\u53d8\u6389\uff0c
\n\\begin{align}
\n\\tilde{x} = \\frac{\\kappa}{\\sqrt{m\\gamma}}x, \
\n\\tilde{t} = \\frac{\\kappa}{m}t.
\n\\end{align}
\n\u5e26\u5165\u539f\u65b9\u7a0b\u5e76\u4e14\u8003\u8651\u5230\(y=\\frac{dx}{dt} = \\sqrt{\\frac{\\gamma}{m}}\\tilde{y}\)\uff0c\u5f97\u5230
\n\\begin{align}
\n\\dot{\\tilde{x}} = \\tilde{y}, \
\n\\dot{\\tilde{y}} = -\\tilde{y} + sign\\left(q\\right)\\frac{1}{\\tilde{x}}.
\n\\end{align}
\n\u6211\u4eec\u770b\u5230\u9664\u4e86\(q\)\u7684\u7b26\u53f7\u4f1a\u8fdb\u5165\u65b9\u7a0b\uff0c\u5269\u4e0b\u7684\u6240\u6709\u7684\u5e38\u6570\u90fd\u6d88\u5931\u4e86\uff0c\u4e5f\u5c31\u662f\u90fd\u7edf\u4e00\u5230\u4e86\u4e00\u4e2a\u65b9\u7a0b\uff0c\u4e00\u79cd\u884c\u4e3a\uff0c\u800c\u4e0d\u4f9d\u8d56\u4e8e\u53c2\u6570\u503c\u3002\u8fd9\u662f\u5f88\u597d\u7684\u6027\u8d28\u3002\u5f53\(q>0\)\u7684\u65f6\u5019\uff0c\u8fd9\u4e2a\u65b9\u7a0b\u6b63\u597d\u5c31\u662f\u5bf9\u5e94\u7740\u524d\u9762\(a=1,b=1\)\u7684\u60c5\u5f62\u3002\u4e8e\u662f\uff0c\u8fd9\u5c31\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u4e0d\u7ba1\u53c2\u6570\u5982\u4f55\u9009\uff0c\u53ea\u8981\(q>0\)\uff0c\u6570\u503c\u8ba1\u7b97\u51fa\u6765\u7684\u66f2\u7ebf\u7684\u6837\u5b50\u90fd\u4e00\u6837\u3002<\/p>\n

\u786e\u5b9e\u80fd\u591f\u770b\u5f97\u66f4\u52a0\u6df1\u523b\u3002\u91cf\u7eb2\u5206\u6790\u8fd8\u662f\u4e0d\u9519\u7684\u4e1c\u897f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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