The critical points of the elastic energy among curves pinned at endpoints

Figure 1.  Critical points of $ \mathcal{W} $ in $ \mathcal{A}_{l, L} $ are given by Theorem 1.1. According to [41], these curves are called wavelike elasticae

Figure 2.  The relation between critical curves and the ratio $ l/L $. The number of inflection points (where the sign of the curvature changes) in $ (0, l) $ is given by $ n\in \mathbb{N}\cup\{0\} $. The curve $ \hat{ \gamma}^{\pm}_n $ ($ n\in \mathbb{N} $) can be constructed from $ \hat{ \gamma}^{\pm}_0 $.

Figure 3.  For any $ n\in \mathbb{N}\cup\{0\} $ the curves $ \check{ \gamma}^+_n $ and $ \check{ \gamma}^-_n $ has a loop

Figure 4.  The red curve is the graph of $ 2\frac{ \mathrm{E}(p)}{ \mathrm{K}(p)}-1 $; $ 0\leq p \leq p_0 $ and $ -2\frac{ \mathrm{E}(p)}{ \mathrm{K}(p)}+1 $; $ p_0< p <1 $