# Characterization of the critical points for the free energy of a Cosserat problem

## TL;DR

This paper explicitly computes critical points of the free energy in a Cosserat model using the embedded gradient vector field method, providing conditions for critical points and addressing an open question on local versus global minima.

## Contribution

It introduces explicit formulas for critical points and Hessians in Cosserat models and answers an open question about the nature of local and global minima.

## Key findings

- Explicit critical points for Cosserat free energy are derived.
- Necessary and sufficient conditions for critical points on SO(n) are formulated.
- It is shown that all local minima are global minima in this context.

## Abstract

Using the embedded gradient vector field method we explicitly compute the list of critical points of the free energy for a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group 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 in L. Borisov, A. Fischle, P. Neff, "Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices", ZAMM (2019), namely if all local minima of the optimization problem are global minima. We point out a few examples with physical relevance, in contrast to some theoretical (mathematical) situations that do not hold such a relevance.