Post-Newtonian N-Body Dynamics in Extended Theories of Gravity

We derive the complete first post-Newtonian (1PN) Lagrangian and corresponding equations of motion for the relativistic $N$-body system in Scalar-Tensor-Fourth-Order Gravity (STFOG), including the Non-Commutative Spectral Geometry (NCSG) sector as a special case. In the regime $\Phi \sim \Psi$ ($\gamma \sim 1$), the linearized fourth-order field equations are solved in the Standard Post-Newtonian gauge, and the variational Lagrangian is built directly from the point-particle action. The resulting dynamics is governed by three Yukawa functions $\zeta$, $\mathcal{W}$ and $\Xi$, which encode the scalar, gravitomagnetic and three-body sectors and depend on the effective masses $(m_R,m_Y,m_\phi)$ of the additional propagating modes. In this context, we show that the nonlinear metric component ${}^{(4)}\!g_{00}$ plays no role at 1PN level. The 1PN orbital motion of the above extended theories is thus obtained in closed form, and the Einstein--Infeld--Hoffmann equations are recovered in the corresponding general-relativistic limit. The formalism provides a common framework for the relativistic celestial mechanics of the Solar System, binary pulsars such as PSR J0737-3039, Galactic-center stellar orbits and triple systems.