Perhaps it is a simple question, but I am unable to solve it. Let us suppose that we have a body confined in $\textbf{R}^{3}$ whose mathematical description is given by a bounded and closed domain $\textbf{D}\subset\textbf{R}^{3}$. Moreover, let us also suppose that its mass distribution is given by the function $m:\textbf{D}\rightarrow\textbf{R}_{\geq0}$ such that \begin{align*} \int_{\textbf{D}}m(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z = M > 0 \end{align*}
My question is: given that its center of mass' trajectory is described by the curve $\alpha:[0,1]\rightarrow\textbf{R}^{3}$, how do we describe the mass distribution $m(x,y,z,t)$ along time?
It is worth mentioning here that we are not considering any kind of rotation, and we are dealing with a rigid body.