Um estudo de dinâmica na mecânica analítica e o formalismo de “Eisenhart Lift”

Abstract

This work revisits the dynamics of conservative systems from a geometric viewpoint. Starting with the Newtonian framework and the weak/slow limit of General Relativity, we show how the Newtonian acceleration ẍ = −∇Φ, with Φ = −GM/r, arises from relativistic geodesics when g₀₀ ≃ −(1 + 2Φ/c²). In this setting, we employ the Eisenhart lift to reinterpret the evolution of a particle—originally governed by a Lagrangian with a potential—as a geodesic on a manifold endowed with one extra coordinate. This reformulation yields, in the effective potential of the central problem, a corrective term +C/(2r²) that shifts the radial frequency and induces a mild periapsis precession, while recovering the Kepler case when the correction is turned off (C = 0).

We derive the Euler–Lagrange equations step by step, identify the first integrals (angular momentum and the constant associated with the extra coordinate), build the effective energy, and obtain Binet’s equation, from which we get the analytical prediction for both the periapsis precession and the near-sinusoidal form of r(t) around nearly circular orbits. Numerically consistent integrations display a clean r(t) and orbits with the expected precession for C > 0, in agreement with the analytical expressions. The formalism is compatible with the Einsteinian weak-field limit adopted here and provides a geometric control parameter for orbital corrections without departing from the underlying classical physics.