The advantage of using a computer to find a numerical solution rather than an analytical solution is quite clear when considering the relationship between Jupiter, the Sun, and the Earth. Rather, it is impossible to achieve an analytical solution for most celestial mechanics problems involving 3 or more bodies, thus numerical approaches are the only alternative.
Our project begins with modeling the Sun/Earth/Jupiter system with the Sun fixed at the origin. The three bodies are related via the inverse square law, and the positions of Jupiter and the Earth are calculated using their respective equations of motion (the sum of the forces from the other two bodies). Once a working model of the planetary orbits is established, it is worth investigating the effects of Jupiter on the Earth’s orbit when Jupiter’s mass is increased. We know that the current orbit is stable, but at what point does it become unstable?
If the mass of Jupiter is increased by roughly a thousand fold, it actually becomes comparable to the Sun’s mass. In this case, the model can be expanded to approximate a true 3-body system, with the origin at the center of mass of the system instead of fixed at the sun. The advantage of this model is that it allows us to observe how all three planets are effected by this change in mass. How large an effect, and at what point the orbits becomes unstable, remain open to investigation.
In your first paragraph you talk about the relationship between three celestial bodies. What kind of physical relationship are you thinking of? Are you describing this relationship kinetically or through gravitational fields? What is your goal?