= Acceleration potential of a homogeneous motion
{title2=$\ddot F F^{-1}=(\ddot F F^{-1})^T$}
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 $\xi=-\mathbf y^TA\mathbf y/2-\ddot{\mathbf c}\cdot\mathbf y+g(t)$. Direct spatial differentiation gives $-\nabla\xi=A\mathbf y+\ddot{\mathbf c}$. Equivalently $F^T\dot F-\dot F^TF$ is constant: its time derivative is $F^T\ddot F-\ddot F^TF$.
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