Stability results for constrained calculus of variations problems: an analysis of the twisted elastic loop
Stability results for constrained calculus of variations problems: an analysis of the twisted elastic loop
复制标题
约束变分问题的稳定性结果：扭曲弹性环的分析
DOI：
10.1098/rspa.2004.1435
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Kathleen A. Hoffman
中科院分区：
文献类型：
--
作者：
Kathleen A. Hoffman
变分结构的问题在整个物理科学中无处不在，并有着杰出的科学史。约束变分问题的研究要少得多，特别是稳定性理论，它决定了哪些解是物理上可实现的。在这篇文章中，我们发展了适用于参数相关变分问题的稳定性交换结果，该问题具有两个特殊的特征：要么参数出现在边界条件中，要么存在等周约束。特别地，我们确定了一个相关的区分分岔图，它编码了褶皱稳定性交换的方向。我们将该理论应用于扭曲的弹性环，它可以自然地表示为具有等周约束和参数相关边界条件的变分问题。与将某些干草分叉分类为亚临界或超临界的扰动展开相结合，扭曲回路的区别图提供了对所有平衡点的稳定性属性的分类。特别是，揭示了稳定性能与扭转与弯曲刚度之比的出乎意料的敏感依赖性。
Problems with a variational structure are ubiquitous throughout the physical sciences and have a distinguished scientific history. Constrained variational problems have been much less studied, particularly the theory of stability, which determines which solutions are physically realizable. In this paper, we develop stability exchange results appropriate for parameter-dependent calculus of variations problems with two particular features: either the parameter appears in the boundary conditions, or there are isoperimetric constraints. In particular, we identify an associated distinguished bifurcation diagram, which encodes the direction of stability exchange at folds. We apply the theory to a twisted elastic loop, which can naturally be formulated as a calculus of variations problem with both isoperimetric constraints and parameter-dependent boundary conditions. In combination with a perturbation expansion that classifies certain pitchfork bifurcations as sub- or super-critical, the distinguished diagram for the twisted loop provides a classification of the stability properties of all equilibria. In particular, an unanticipated sensitive dependence of stability properties on the ratio of twisting to bending stiffness is revealed.
影响因子：
6.8
作者：
Olson, WK
通讯作者：
Olson, WK
DOI：
10.1103/physreve.61.747
发表时间：
2000
期刊：
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
影响因子：
--
作者：
I. Tobias;D. Swigon;B. D. Coleman
通讯作者：
I. Tobias;D. Swigon;B. D. Coleman