Consult the project "Moving stars around", there is an old C and a modernized Ruby version. (And now a C++ version?)
Short advice: Eulers methods are bad for energy conservation. explicit Euler increases energy, implicit Euler reduces energy. Just check it on the phase space picture of the harmonic oscillator y''+y=0.
Use symplectic integrators, the most simple and famous is the Leapfrog or Verlet method that already Newton used to reason about planetary movement.
