Acceleration potential of a homogeneous motion ($\ddot{F}{F}^{-1}=(\ddot{F}{F}^{-1}{)}^T$)

A homogeneous motion has an acceleration potential exactly when $A=\ddot{F}{F}^{-1}$ is a symmetric matrix. Necessity follows because $A$ is the spatial gradient of the material acceleration, which must be a negative Hessian matrix. For sufficiency, put $\mathbf{y}=\mathbf{x}-\mathbf{c}(t)$ and take $\xi =-{\mathbf{y}}^{T}A\mathbf{y}/2-\ddot{\mathbf{c}}\cdot \mathbf{y}+g(t)$. Direct spatial differentiation gives $-\nabla \xi =A\mathbf{y}+\ddot{\mathbf{c}}$. Equivalently $F^T\dot{F}-{\dot{F}}^{T}F$ is constant: its time derivative is $F^T\ddot{F}-{\ddot{F}}^{T}F$.