Is is possible to use Runge-Kutta RK4 on the three body problem? I have just learned about the RK4 method for approximating differential equations. Is it possible for this to be applied to the three-body problem?
 A: Yes, it is possible. It is an ODE system, so you can apply numerical methods for ODE.
What you should be asking about is the quality of the numerical approximation. For one, in the long term the method step errors will accumulate and distort the physical picture. 
In the short term, many instances of the 3 body problem will have phases when two bodies come close together. In those time segments the system is very stiff, requiring a very small step size. A very short step size in a fixed-step method like RK4 implies a rapid build-up of floating-point errors, limiting the useful integration interval.
For the preservation of physical properties you need to use symplectic methods, as they preserve constants such as energy, momentum and angular momentum to a higher order than that of the method. Alternatively, implicit methods are more robust towards the stiff phases, requiring much less step size reduction. In a next step, operator splitting method such as exponential methods will also allow slightly longer step sizes.
