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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 $F^T\ddot{F}-{\ddot{F}}^{T}F$.

Let ${\mathbf{x}}_0(\mathbf{X};\tau )$ be a family of static elastic equilibrium solutions for a homogeneous material with incompressibility under a fixed spatial body force. Replacing $\tau$ by time gives a dynamic solution of the elastic field equations exactly when the resulting material acceleration has a global acceleration potential $\xi$. At the same deformation gradient, the Cauchy stress tensor can change only by the scalar pressure reaction. Subtracting static and dynamic balance of linear momentum gives $\nabla q=\rho \mathbf{a}$, and constant mass density then gives $q=-\rho \xi +C(t)$. Surface tractions must match the adjusted Cauchy stress tensor.

Restore the particle label suppressed in the question, writing the static family as ${\mathbf{x}}_0(\mathbf{X};\tau )$. At each fixed $\tau$, its Cauchy stress tensor satisfiesThis dynamic continuation of a static incompressible elastic deformation sets $\tau =t$ to form a motion. At every time its deformation gradient is the same as that of the corresponding static member. For an elastic material the elastic part of the Cauchy stress tensor depends on that deformation gradient, while incompressibility leaves an arbitrary scalar constraint reaction in hyperelastic stress. Therefore writeHere $1$ is the identity matrix. Subtracting static balance of linear momentum from dynamic balance of linear momentum givesSince the homogeneous incompressible material has constant mass density, define $\xi =(p-p_0)/\rho$. NecessarilyConversely, suppose this acceleration potential exists. Choose $p=p_0+\rho \xi$. ThenThe added isotropic stress is symmetric, so angular momentum conservation remains satisfied. This proves both directions for the field equations and the elastic constitutive equation. The surface tractions for the motion are those of the adjusted stress; identical prescribed static tractions would impose an additional boundary restriction on $\xi$.