How do you calculate unified planetary precession incl. both GR and Newtonian contributions?

Question

We know the theory of general relativity (GR) accounts for the illusive $43$ arc seconds of precession of the perihelion of Mercury. The calculation is well known and well studied. How does one account for the remaining planetary precession? So, more specifically, the effect that the other planets have on the precession of Mercury. I know there are methods listed in the paper by Price and Rush, 1978, which states there is more methods presented in undergraduate mechanics books. However, I have never came across such things.

Is there a method in which one can derive both the planetary and GR precession effects or do we need to separate the precession into a Newtonian and relativistic form and then add them together?

EDIT: Furthermore, what about an arbitrarily chosen planet? The ring method mentioned in the answer by @diracology and the paper that I have linked assumes assumes a uniformly distributed ring of mass surrounding Mercury. However, what if we were concerned with planetary and GR precession effects of Jupiter (just as an example) - Do we then proceed to consider two rings around Jupiter?

Answer 1

It is possible to derive the precession of Mercury perihelion in some sort of unified way. However we still need to take into account the Newtonian and Relativistic contributions. They cannot be put in equal foot from the physical point of view.

The idea is to write the effective potential with the centrifugal term, the non-perturbative term that represents the interaction between the Mercury and the Sun, the Newtonian gravitational interaction with the other planets and the Relativistic correction due to spacetime deformations. The last two contributions being small perturbative terms.

The idea is that since this effective potential has a stable minimum, it can be Taylor expanded. The second order term actually gives the period of the radial oscillations from which we can obtain the angular displacement (for small perturbations). If the angular displacement corresponding to one period is different to $2\pi$ it means that the planet's orbit is precessioning.

The complete derivation can be seen in this post Details of Newtonian Prediction for Mercury's Precession. You only need to take into account the relativistic contribution $$U_{GR}=-\frac{GML^2}{mc^2r^3},$$ where $G$, $M$, $c$, $r$, $m$ and $L$ are the Universal constant, Sun's mass, the speed of light, the distance from the Sun to the planet, the mass and the angular momentum of the planet, respectively. The two terms together give a total of $575$ arcsecond per century.