Orbits in Schwartzschild spacetime
Assuming that the orbits remain permanently in the equatorial plane [ as in Newtonian theory ] i.e. , the Lagrangian is:
The Euler- Lagrange equations are
This is just energy conservation. Also
This is conservation of angular momentum.
Remember that with for timelike orbits and for null orbits, so
Substituting these into the above, we get
This is the Newtonian energy equation with a modification to the term. Using and putting we get
This equation can in fact be integrated immediately, but it leads to elliptical integrals, which are awkward to handle. We therefor differentiate to obtain the equation:
For timelike orbits [ ] this is just Newton's equation
apart from the last term.