论文标题
异质Helicoseir
The heterogeneous helicoseir
论文作者
论文摘要
我们在重力作用下研究了一个自由端点，并具有任意密度的垂直轴的重弦（Helicoseir）的旋转。我们表明，如在统一情况下，问题可以转化为非线性特征值方程。该方程式的本征模表示旋转字符串的平衡配置，其中字符串的形状不会随时间而变化。正如Kolodner先前证明的那样，非线性方程式的新模式的出现与相应的线性方程的光谱相关。我们已经能够将此结果概括为一类密度$ρ（s）=γ（1-s）^{γ-1} $，其中包括同质字符串作为特殊情况（$γ= 1 $）。 我们还表明，任意密度的非线性特征值方程（NLE）的解是正交的，并且该方程的解决方案具有给定数量的节点的解决方案包含不同的Helicoseir的溶液，具有较小的节点。这两种属性也适用于同质案例，并且以前尚未建立。
We study the rotations of a heavy string (helicoseir) about a vertical axis with one free endpoint and with arbitrary density, under the action of the gravitational force. We show that the problem can be transformed into a nonlinear eigenvalue equation, as in the uniform case. The eigenmodes of this equation represent equilibrium configurations of the rotating string in which the shape of the string doesn't change with time. As previously proved by Kolodner for the homogenous case, the occurrence of new modes of the nonlinear equation is tied to the spectrum of the corresponding linear equation. We have been able to generalize this result to a class of densities $ρ(s) = γ(1-s)^{γ-1}$, which includes the homogenous string as a special case ($γ=1$). We also show that the solutions to the nonlinear eigenvalue equation (NLE) for an arbitrary density are orthogonal and that a solution of this equation with a given number of nodes contains solutions of a different helicoseir, with a smaller number of nodes. Both properties hold also for the homogeneous case and had not been established before.
