Theory and Modern Applications
From: Conditional linearization of the quintic nonlinear beam equation
Group | Generators | Ordinary differential equations |
---|---|---|
1 | \(X_{1}+X_{2}\), \(X_{1}+X_{2}+X_{3}\) | \(\begin{array}[t]{l} c\phi'+(m+P_{0})\phi''+EI\phi^{(4)}-3EI\phi ^{\prime\prime 3}+\frac{3}{2}P_{0}\phi^{\prime 2}\phi''\\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi^{\prime\prime 3}+\frac {9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \) |
\(X_{1}+X_{2}+X_{4}\) | \(\begin{array}[t]{l} c+c\phi'+(m+P_{0})\phi''+EI\phi^{(4)}-3EI\phi ^{\prime\prime 3}+\frac{3}{2}P_{0}\phi^{\prime 2}\phi'' \\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi^{\prime\prime 3}+\frac {9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \) | |
2 | \(X_{2}+X_{4}\), \(X_{2}+X_{3}+X_{4}\) | \(\begin{array}[t]{l} c+P_{0}\phi''+EI\phi^{(4)}-3EI\phi^{\prime\prime 3}+\frac {3}{2}P_{0}\phi^{\prime 2}\phi''\\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi^{\prime\prime 3}+\frac {9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \) |
\(X_{2}+X_{3}\) | \(\begin{array}[t]{l}P_{0}\phi''+EI\phi^{(4)}-3EI\phi^{\prime\prime 3}+\frac {3}{2}P_{0}\phi^{\prime 2}\phi''\\ \quad{} -3EI\phi^{\prime 2}\phi^{(4)}+\frac{27}{2}EI\phi^{\prime 2}\phi ^{\prime\prime 3}+\frac{9}{4}EI\phi^{\prime 4}\phi^{(4)}=0 \end{array} \) |