(A) In a three-dimensional flat space, total energy $H = T + U$ in a conservative field is constant. The three-dimensional Lagrangian ($L = T - U$) gives force on an object (see image- eq. (1)). $T$ is kinetic energy (K.E.) and $U$ is potential energy (P.E.).
We can also write the potential energy in kinetic energy form by introducing an additional coordinate u to write the four-dimensional total energy equation. It has to be presumed that all the four coordinates satisfies the basis vector symmetry, which is the requirement of the Lagrangian analysis.
As the classical potential is written in kinetic energy form and the total energy equation is four-dimensional with only kinetic energy terms, the resultant system force is zero (eq. (3) in image). But, both the eq. (1) and (3) give same results for force on the individual mass.
(B) We can multiply this total energy with a function and rearrange it to write the modified total energy equation. We have to choose an appropriate function to get the desired results. The three dimensional Lagrangian eq. (1) then gives modified force values for the modified kinetic energy and potential. The four-dimensional eq. (3) will also give same force for the modified total energy equation.
We can modify the total energy equation to get same results for the planetary orbits as the Schwarzschild metric results in geodesic analysis. Mathematical analysis of the classical Lagrangian approach and the GR approach is very similar, but physical interpretations are different. Only for the limited purpose of obtaining same planetary orbits, we can establish correspondence between the two approaches as:
Compared with the classical picture, s in GR represents time and the fourth coordinate t in GR represents the classical potential U.
(C) But, this modification changes the basis vectors and unit vectors. It can be difficult to write the expressions of unit vectors if there is no geometry to justify basis vectors symmetry. Then it will be difficult to even draw a coordinate system to describe motion of a body.
The question is: This classical Lagrangian analysis with the modified energies, is mathematically similar to the geodesic equations analysis for the four-dimensional curved space Schwarzschild metric. The results for planetary orbits are same. Then, does it mean that, we need a modified expression of the kinetic energy for a mass in motion under gravity?
