We have a system of $n$ particles, and the particle $i$ has a point mass $m_i$. The center of mass is then given by:
$X = \frac{\sum_i^nm_ix_i}{\sum_i^nm_i}$ $Y = \frac{\sum_i^nm_iy_i}{\sum_i^nm_i}$ $Z = \frac{\sum_i^nm_iz_i}{\sum_i^nm_i}$
Now imagine that we want to add a specific amount of extra mass $M_a$, to the existing particle system, in order to guide the center of gravity to a new position, X', Y', Z'. Taking into account that the number of particles can be a large number (e.g 1M), is there an algorithm (iterative or not) that I can use to find how to distribute the mass to the existing particles? The whole amount of $M_a$ must be used, and the initial mass $m_i$ of the $i$-th particle cannot be changed.
I'm thinking of starting with each particle having an equal amount of extra mass, e.g $M_a / n$, and then, through some iterative process (which I still have to figure out), guide the center of mass to the desired position.