Equations of motion for Earth and Moon One of the simplest versions of Newton's equations of motion for the Sun, Earth and Moon can be obtained by making the approximation that the three bodies are perfect spheres. In this approximation, the pair potential between body i and j is  $$\frac{-G m_i m_j}{ r_{ij}},$$ where $r_{ij}$ is the distance between the bodies: one can solve the equations numerically and get the orbits. 

This simple model does not include multiple, additional factors, such as 


the fact that the Earth is not perfectly spherical (equatorial bulge),
Earth's tides,
the influence of other celestial bodies,


and others.

Amongst all these factors, what is the dominant one as for predicting the orbit of the Earth? In other words, what is the leading factor that should be included in the simple model above in order make its predictions for the Earth's orbit closer to the observations?
I may have found a partial answer to my question in the Navigational Astronomy chapter of 'American Practical Navigator' by N. Bowditch . On page 228 


  [...] nutation, an irregularity in the motion of the Earth due to the
  disturbing effect of other celestial bodies, principally the Moon


and on page 230 


  The precession of the Earth’s axis is the result of gravitational
  forces exerted principally by the Sun and Moon on the Earth’s
  equatorial bulge. The spinning Earth responds to these forces in the
  manner of a gyroscope. Regression of the nodes introduces certain
  irregularities known as nutation in the precessional motion.


So the dominant effects as for predicting the orientation of the Earth's axis seem to be the Earth's ellipticity combined with the attraction of the Sun and Moon.