General Relativity: An Introduction for Physicists · Problem 9.10
The Schwarzschild geometry — Hobson, Efstathiou & Lasenby
Physics · constants-of-motion-planar-orbits
Problem
Show that the quantity
L2=pθ2+sin2θpϕ2,
where p is the 4-momentum of a particle, is a constant of motion along any geodesic in the Schwarzschild geometry. Hence show that the particle orbits in a Schwarzschild geometry are stably planar.