The N-body problem is the problem of finding how N-masses interact gravitationally. Although it has no closed form solution, the solution can be approximated using Maple.

To understand the problem better, we solve first the reduced 3-body problem, which is the same problem with N=3 and some of the masses being stationary. A fairly good exercise for the mechanics of this problem, is to simulate sending a probe to the Moon from Earth. In this case we have 2 stationary bodies (Earth and Moon) and we seek the calculational details which will allow us to put the probe in lunar orbit.

Let's assume that our probe has a mass of 1000 tons. The whole procedure consists of three subproblems:

1. Vertically launch to altitude ae, after which,
2. Put the probe in Earth orbit.
3. Shoot the probe towards the Moon.
4. Put the probe in Lunar orbit.

The first step is covered in most standard ballistics texts. The next three steps require a fairly more elaborate understanding.

Geodesics

First we need to understand the geodesics of gravitational attraction around the system^[1]. Once we have the geodesics of the system, we can visualize the gravitational pool that's created around the Earth and Moon system. The geodesics are given by Newton's Law of Universal Gravitation (NLUG).

Earth-Moon unit-distance system geodesics (cyan) and gravitational pool (purple/blue). Masses are indicated relative to ME=1

Earth Orbit

The geodesics which characterize a 1000 ton probe in medium orbit are shown:

Earth-probe geodesics and gravitational pool for Medium Earth orbit

To put the probe in Earth orbit, we find a velocity which keeps it roughly on an elliptical orbit. For this we choose an altitude and try to find a velocity which keeps it in orbit. For our example, we choose an altitude of ae=6000 km, which gives a medium orbit. For such an orbit, a velocity u=2.85 km/sec suffices:

Probe in Medium Earth orbit

Lunar Orbit

The geodesics which characterize a 1000 ton probe in lunar orbit are shown:

Moon-probe geodesics and gravitational pool for Lunar orbit

To put the probe in Lunar orbit, we again choose a desired altitude and a velocity which keeps the probe in orbit. For our example, we choose an altitude of am=1200 km, in which case a velocity u=0.8 km/sec suffices:

Probe in Lunar orbit

Earth-Moon Trajectory

The geodesics which characterize a 1000 ton probe thrown towards the Moon are shown:

Earth-Moon-probe actual-distance system geodesics (cyan) and gravitational pool (brown/orange). Masses are indicated relative to ME=1

The in between trajectory can be calculated now. After one full revolution from the Earth orbit, we fire our thrusters and give the probe some appropriate velocity to send it to the moon. How do we calculate the appropriate velocity to give to the probe? Let's assume that our starting location is at θ=-3*π/4 around Earth and we need to reach the destination position -3*π/4 around the moon. We would reach this position if we started with an escape velocity ve equal to infinity. Because we can't do that, we need to reduce the velocity ve to ve', one which will bring us am-close to the moon, where am is the moon altitude of the lunar orbit we need to go into. After testing drag and some trial and error, we find that an escape velocity of ve~14.18 km/sec is sufficient^[2].

Probe in transit to Moon

Now if we combine the three steps in the correct order, we can simulate Apollo 11's travel to the moon^[3][4].

A different more distant orbit for the reduced 3-body problem is shown below:

Probe in distant orbit around Earth

An animated version of the above orbit is shown below:

Probe in distant orbit around Earth

The General N-body Problem

The general N-body problem involves finding all trajectories of N interacting masses, under the influence of their own gravity as a function of time. For this, one sums all forces involved and allows a minimum dt which determines the new positions of all masses. It is a bit tricky to figure out the velocities of the masses so that the system is in a stable state, but can be done.

Solution for 4-body problem

An example is given on the figure above, which shows the partial trajectory for 4 masses under the influence of gravity, an Earth and three moons, one with mass equal to Earth's Moon and two more moons with masses 2 and 3 times that of Earth's moon, with all masses approximately in orbit around the Earth mass^[5].

Notes

1. Note that because NLUG has the form F=G*m₁*m₂/r², the geodesics of any gravitational system are described exactly by the complex function f(z)=1/z². For example, the unit distance geodesics and gravity pool for a system of two masses equal to Earth are visualized using f(z) as follows:
   
   Unit distance geodesics and gravity pool for two Earths^[3].

   Note the vertical lines where the space-time continuum tears because of gravity. These tears tend to create a 2d singularity in space-time.

2. Note that this velocity is naturally greater than the escape velocity of Earth at altitude ae, which is ve~8 km/sc.
3. Download a Maple 9 worksheet which shows the calculations for the Geodesics of two Earths, here.
4. Note that the solution described is only an approximation, since by the time the probe reaches the desired destination, the Moon will have moved some. To deal with the latter, one needs to solve the general N=3-body problem numerically, or, continuously adjust the trajectory of the probe to compensate for the Moon's movement.
5. Download a Maple 9 worksheet which shows the complete trajectories of all 4 masses as an animation, here. Note that the general dynamics of the gravitational trajectories for the N-body problem are in general chaotic, so if you make changes to the velocities don't expect nice orbits. The velocities of the four bodies in the simulation were chosen so that all the orbits are almost-stable. It requires considerable effort to put N-bodies in almost-stable orbits around each other. A fairly good exercise for the reader would be to change the positions and velocities of the orbiting bodies in the Maple worksheet and subsequently try to adjust the velocities so that all the orbits are almost-stable. This will make the reader appreciate how much effort has been put forth to create a long-range, dynamically stable system, such as the Solar System.