Many physical systems can be modelled as parameter-dependent variational problems. The associated equilibria may or may not exist realistically and can only be determined after examining their stability. Hence, it is crucial to determine the stability and track their transitions. Generally, the stability characteristics of the equilibria change near folds in the parameter space. The direction of stability changes is embedded in a specific projection of the solutions, known as distinguished bifurcation diagrams. In this article, we identify such projections for variational problems characterized by fixed-free ends -- a class of problems frequently encountered in mechanics. Using these diagrams, we study an Elastica subject to an end load applied through a rigid lever arm. Several instances of snap-back instability are reported, along with their dependence on system parameters through numerical examples. These findings have potential applications in the design of soft robot arms and other actuator designs.

翻译：许多物理系统可建模为参数依赖的变分问题。相关平衡态在现实中可能存在也可能不存在，且必须通过稳定性检验才能确定。因此，判定稳定性并追踪其转变过程至关重要。一般而言，平衡态的稳定性特征会在参数空间的褶皱附近发生变化。稳定性变化的方向蕴含在解空间的特定投影中，这种投影被称为特化分岔图。本文针对具有固定-自由端特征的变分问题——这是力学中常见的一类问题——识别了此类投影。利用这些分岔图，我们研究了通过刚性杠杆臂施加端部载荷的弹性杆问题。通过数值算例，报告了若干回弹失稳现象及其对系统参数的依赖关系。这些发现对软体机械臂及其他致动器设计具有潜在应用价值。