Mercury’s Perihelion Precession in a Unified Description of Gravity and Motion: A Hamiltonian Approach

This work investigates the perihelion advance of Mercury in a generalized description of gravity in which the speed of light depends on the cumulative gravitational potential of the Universe. Starting from a relativistic Lagrangian with c = c(Φ), the orbital equations are derived including the effect of the retarded gravitational potential. The resulting precession separates into intrinsic and retarded-potential contributions. For the particular Lagrangian formulation adopted, their combination reproduces the first-order General Relativity prediction. A generalized two-parameter family of Lagrangians is then introduced to examine the dependence of the precession on the underlying Lagrangian structure. While the adopted formulation reproduces the observed perihelion advance, alternative structures considered within this framework do not straightforwardly reproduce the observed value.