Sophus Lie等谱Euler-Bernoulli梁
iso-spectral Euler-Bernoulli beams à la Sophus Lie
摘要：
利用因子分解和Lie对称技巧得到了等谱Euler-Bernoulli光束。将规范Euler-Bernoulli梁算子分解为二阶线性微分算子与其伴随算子的乘积。然后将因子反转以获得等谱光束。如果因子的系数满足非线性常微分方程系统，则因子分解是可能的。该系统的解耦产生一个单一的非线性三阶常微分方程。用李群方法分析和求解了这个常微分方程，称之为{ \it主方程}。证明了主方程可以是一维或三维对称李代数。当主系统具有唯一的对称性时，我们所能做的最好的事情就是把它的阶数降低一阶。在这种情况下，我们得到了一个单参数解族。最大对称的情况下，示出同构的Chazy方程，其在封闭形式中求解，以导出主方程的一般解。
英文摘要：
We obtain iso-spectral Euler-Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain iso-spectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations . The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, refer to as the {\it principal equation}, is analyzed and solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation.
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