General Relativity: An Introduction for Physicists · Problem 3.17

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · geodesic-euler-lagrange-lagrangian

Problem

By substituting the 'Lagrangian'

L = g_{ab}\dot{x}^a \dot{x}^b

into the Euler–Lagrange equations, show directly that

\ddot{x}^a + \Gamma^a{}_{bc}\dot{x}^b \dot{x}^c = 0,

where the dots denote differentiation with respect to an affine parameter.