We can see the interaction term LI 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." And 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. This is done without quoting any reference. Will anybody provide me with any authentic standard reference to this derivation?
The text is too vague to be of any use. It's not a good idea to use such a source for a technical subject that is by now standard material in any physics curriculum.
The Lagrangian of a point is given by the integral of the proper time in soacetime. The proper time in a general background is given by the expression
ds=(gμν (dxμ/dλ)(dxν/dλ))1/2 dλ
Here λ parametrizes the worldline of the particle in spacetime. The integral of this expression over the parameter λ is invariant under reparametrization of the worldline, since a reparametrization, λ=f(μ), produces a Jacobian, which cancels the change of variables in ds.
Dimensional analysis imposes that the classical action is given by the integral of
-mc ds
where c=speed of light in vacuum and a standard exercise is to show that m can be identified with the mass of the particle (invariance under reparzametrization is a gauge symmetry that must be fixed).
This is described quite thoroughly in https://inspirehep.net/literature/4176
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