Using the embedded gradient vector field method, we explicitly compute the set of critical points of the shear-stretch strain energy of a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group \(\text {SO}(n)\). Each critical point is then characterized using an explicit formula for the Hessian operator of a cost function defined on the orthogonal group. We also give a positive answer to an open question posed by Borisov et al. (Z Angew Math Mech 99:e201800120, 2019), namely if all local minima of the optimization problem are global minima.

使用嵌入式梯度矢量场方法，我们显式计算Cosserat体模型的剪切拉伸应变能的临界点集。在特殊正交组\（\ text {SO}（n）\）的抽象情况下，我们还为临界点制定了充要条件。然后，使用针对正交组上定义的成本函数的Hessian运算符的显式公式来表征每个临界点。对于Borisov等人提出的一个悬而未决的问题，我们也给出了肯定的答案。（Z Angew Math Mech 99：e201800120，2019），即优化问题的所有局部最小值是否是全局最小值。