= Solution
Put $\mu=GM/c^2$, $A=1-2\mu/r$ and $B=1+2\mu/r$. For an affinely parametrized <geodesic>, use the unhalved quadratic <Lagrangian> $\mathcal L=-c^2A\dot t^2+B\dot{\mathbf r}^{,2}$. Its <Euler-Lagrange equations> are unchanged by this overall normalization. Since $t$ is a cyclic coordinate, $d(-2c^2A\dot t)/d\tau=0$, equivalently
$$
\boxed{\frac d{d\tau}(2c^2A\dot t)=0.}
$$
The spatial derivatives give $\partial\mathcal L/\partial\dot x^i=2B\dot x^i$ and $\partial\mathcal L/\partial x^i=-c^2\dot t^2A_i+\dot{\mathbf r}^{,2}B_i$, where $A_i=\partial_iA$. Therefore
$$
\boxed{\frac d{d\tau}(2B\dot{\mathbf r})+c^2\dot t^2\nabla A-\dot{\mathbf r}^{,2}\nabla B=0.}
$$
For the <connection coefficients>, take $x^0=ct$, so $g_{00}=-A$ and $g_{ij}=B\delta_{ij}$. Substituting the diagonal <inverse metric> into the <Christoffel symbol> formula gives all nonzero component types, including lower-index symmetry:
$$
\boxed{\begin{aligned}
\Gamma^0{}_{0i}=\Gamma^0{}_{i0}&=\frac{A_i}{2A}=\frac{\mu x_i}{Ar^3},\\
\Gamma^i{}_{00}&=\frac{A_i}{2B}=\frac{\mu x^i}{Br^3},\\
\Gamma^i{}_{jk}&=\frac{\delta^i_kB_j+\delta^i_jB_k-\delta_{jk}B_i}{2B}
=-\frac{\mu}{Br^3}(\delta^i_kx_j+\delta^i_jx_k-\delta_{jk}x^i).
\end{aligned}}
$$
All other component types vanish. If the time coordinate is $t$ rather than $ct$, replace $\Gamma^i{}_{00}$ by $\Gamma^i{}_{tt}=c^2\Gamma^i{}_{00}$; the displayed time–space mixed coefficients are unchanged. Spatial indices in these formulas are raised with the background Euclidean metric.
For the circular orbit let $u=\mu/R$ and $\Omega=d\phi/dt=\omega/\dot t$. Both $A$ and $B$ are constant along it. The radial spatial equation gives $(1+u)\omega^2=(GM/R^3)\dot t^2$, so, treating the displayed metric coefficients algebraically, $\Omega^2=GM/[R^3(1+u)]$. The <weak-field approximation> securely determines the leading <Keplerian orbit> result
$$
\boxed{P_t\simeq2\pi\sqrt{\frac{R^3}{GM}},\qquad P_\tau=\frac{2\pi}{|\omega|},\qquad P_t=\dot t\,P_\tau.}
$$
The time normalization is $-c^2=-c^2A\dot t^2+BR^2\omega^2$, yielding $\dot t=1+3u/2+O(u^2)$ and $P_\tau/P_t=1-3u/2+O(u^2)$. One must distinguish the proper-time angular frequency in the supplied path from the coordinate-time angular frequency. The formal factor $1+u$ in the frequency equation comes from treating the truncated metric as exact; a complete relative post-Newtonian correction to the orbital period would also require the omitted higher-order temporal metric terms.
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