An alternative derivation of the equation of motion in General Relativity

[Bishal Banjara]

TL;DR

I need a standard reference/s for an alternative form of the equation of motion in general relativity that I saw elsewhere.

I am extremely engaged in studying the different ways of derivation of the equation of motion in General Relativity. On the way, I found a very general form of the equation of motion that no standard books have done (to my knowledge).

Although the process is implemented in deriving the equation of motion for charged particles in standard books, it is done by none for the case of mass. Although it is not an authentic reference we can see the interaction term $L_I$ involved (added) to generalize the Lagrangian than that for the free particle somewhere in Wikipedia: https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics under the heading "Lagrangian formulation in general relativity."

This finally leads an extra term to the equation of motion $f_{\alpha}$ representing the additional source of force beside the gravitational force of the mass in reference by varying the lagrangians with the position of the particle. This is done without quoting any reference. I want to know its authenticity. Will anybody provide me with any authentic standard reference to this type of derivation? Or, is this the wrong way to do the job?