"Inverse" $N$-body problem There is a well-known $N$-body problem in classical mechanics: Given an initial positions and velocities of $N$ particles in some space, describe their dynamics over some time interval.
I'm interested in some form of "inverse" problem: Let's assume that we know that there are $(N+1)$ particles in some space. We are given the trajectories of $N$ of these particles over some time interval. The problem is to restore the trajectory of the $(N+1)$-th particle over the same time interval.
The underlying force field is assumed to be known. For example, we can assume that each pair of particles is attracted according to the inverse square law.
What is a correct generally accepted name for this problem? Is this problem described in literature?
 A: If the system is isolated, then the center of mass of this system moves at a constant (usually zero) velocity $\mathbf{v}_c$:
$$
\sum_{i=1}^{N+1}m_i\mathbf{r}_i(t)=M \mathbf{r}_c(t)=M(\mathbf{r}_0+\mathbf{v}_c t)
$$
If $\mathbf{r}_i(t)$ are known for all $i=1,\ldots,N$, then $\mathbf{r}_{N+1}(t)$ can be obtained from this equation:
\begin{equation}\tag{1}
\mathbf{r}_{N+1}(t)=\frac{1}{m_{N+1}}\left(M(\mathbf{r}_0+\mathbf{v}_c t)-\sum_{i=1}^{N}m_i\mathbf{r}_i(t)\right)
\end{equation}
This equation contains 2 unknown parameters: initial position of the center of mass $\mathbf{r}_0$ and its velocity $\mathbf{v}_c$. These parameters can be (presumably) obtained by requiring that the equations of motion hold (since the law of interaction is known).
UPDATE:
To obtain $\mathbf{r}_0$ and $\mathbf{v}_c$ from the equations of motion:
Assume that the potential energy is: $U=\sum_{i=1}^{N}\sum_{k=i+1}^{N+1}U(|\mathbf{r}_i-\mathbf{r}_k|)$. Then the equation of motion for each particle is:
$$
m_i\mathbf{\ddot{r}}_i=-\sum_{k=1}^{N+1}U'(|\mathbf{r}_i-\mathbf{r}_k|)\frac{\mathbf{r}_i-\mathbf{r}_k}{|\mathbf{r}_i-\mathbf{r}_k|}
$$
For the first particle:
$$\tag{2}
m_1\mathbf{\ddot{r}}_1=-\sum_{k=1}^{N}U'(|\mathbf{r}_1-\mathbf{r}_k|)\frac{\mathbf{r}_1-\mathbf{r}_k}{|\mathbf{r}_1-\mathbf{r}_k|}-U'(|\mathbf{r}_1-\mathbf{r}_{N+1}|)\frac{\mathbf{r}_1-\mathbf{r}_{N+1}}{|\mathbf{r}_{1}-\mathbf{r}_{N+1}|}
$$
Substituting solution (1) into equation (2) and setting $t=0$ leads to an equation for $\mathbf{r}_0$. Of course, the equation can be nonlinear and it can have multiple solutions.
After $\mathbf{r}_0$ is found, $\mathbf{v}_c$ can be obtained from the same equation (2) for $t>0$.
