1 - Optical–Mechanical Correspondence and Post-Newtonian Hamiltonians up to the Second Order in Schwarzschild and Kerr Metrics

We present a unified optical–mechanical formulation of post–Newtonian dynamics in static and stationary spacetimes, developed consistently up to second post–Newtonian (2PN) order for Schwarzschild and slow–rotation Kerr geometries. Starting from the constitutive dispersion relation \[E = \frac{mc^2}{1-\beta^2},\] we construct an anisotropic Legendre transform yielding an exact Hamiltonian–Lagrangian pair adapted to direction-dependent limiting velocities. The exact constitutive Hamiltonian is written as \[H = K\,F\!\left(\frac{\Pi}{K^2}\right),\qquad\Pi = c_r^2 p_r^2 + \frac{c_\perp^2}{r^2}L^2,\] where the nonlinear closure function \(F\) is uniquely fixed by the dispersion relation through \[\sqrt{z}=\frac{\beta}{1-\beta^2}+\operatorname{artanh}(\beta),\qquadF(z)=\frac{1}{1-\beta^2(z)}.\] Its low–momentum expansion becomes \[H=K+\frac{\Pi}{4K}-\frac{\Pi^2}{48K^3}+\frac{11\Pi^3}{2880K^5}+O(\Pi^4),\] showing that the quartic coefficient is not an independent phenomenological parameter but follows directly from the constitutive structure. The formalism reproduces the Schwarzschild 1PN Binet equation with the standard relativistic precession term \(3GMu/c^2\), yielding the invariant periapsis advance \[\Delta\varpi_{1\mathrm{PN}}=\frac{6\pi GM}{pc^2}.\] At 2PN order, the monopole sector is retained in a gauge- and closure-aware form rather than reduced to a unique coordinate-independent polynomial set. The extension to the Kerr geometry is formulated through a Randers-type optical structure, naturally generating the expected spin–orbit (\(1.5\mathrm{PN}\)) and spin–spin/quadrupole (\(2\mathrm{PN}\)) contributions. The construction should be interpreted as a constitutive effective theory formulated relative to a preferred medium frame. Lorentz-type relations emerge operationally through reconstructed phase times rather than from fundamental Lorentz covariance. The framework provides a compact nonlinear optical–mechanical route to post–Newtonian orbital dynamics while preserving exact Hamiltonian consistency and correct Newtonian and relativistic weak-field limits.