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Consider a test particle of mass $m$ which is in orbit around a spherical-symmetric body with mass $M$. It therefore has a position as described by the coordinates $r,\phi$, and its motion can be described by the Lagrangian $L$ of the Einstein-Infeld-Hoffman-Equations:

$$L = \frac{mv^2}{2}+ \frac{GmM}{r}+\frac{mv^4}{8c^2} + \frac{3GmMv^2}{2c^2r}-\frac{kmM\left(m+M\right)}{2c^2r^2},$$

where $v$ is the particles velocity.

But the orbit of the test-particle can also be described by the Schwarzschild-Metric and the corresponding Lagrangian $\mathcal{L}$

$$\mathcal{L} = -\frac{1}{2}\left[-\left(1-\frac{2 G m}{c^2 r}\right) c^2 \dot{t}^2 + \left(1-\frac{2 G m}{c^2 r}\right)^{-1}\dot{r}^2 + r^2 \dot{\varphi^2}\right].$$ Where the dot denotes the derivative with respect to the proper time of the particle $\tau$ along the world line.

I know that the Newtonian Lagrangian for a testparticle can be derived by requiring $\frac{v}{c}\rightarrow 0$.

Since $L$ simply adds some extra terms to the Lagrangian, it should be possible to do something similar here.

But what kind of expansion is needed to arrive at $L$ from $\mathcal{L}$?

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