Lagrangian Solution of Schwarzschild-like Metric for an ‎Elliptical Object

Abstract

Lagrangian method applied as well as tensor method, for a linear transformed geodesic line element of Schwarzschild-like The ‎Lagrangian method was applied for a linearly transformed geodesic line element of a Schwarzschild-like solution instead of ‎the tensor method. The solution shows that it is not only valid for spherical objects but also it is more comprehensive for ‎elliptical celestial objects. Two types of kinetic and potential energy are the basis of the calculation. Hamiltonian and ‎Lagrangian equality show that the problem has no potential energy. With this transformed geodesic line element, we obtained ‎a new coefficient for the meridional advance of an experimental particle in Schwarzschild spacetime in terms of period, ‎eccentricity, and mean distance. This new perigee equation is not only valid for the Schwarzschild metric (for a spherical ‎object), but also more accurate for the Schwarzschild-like metric (for elliptical objects).‎