Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 64 / 1 / i / Solution

The spatial coordinates are cyclic because the flat Friedmann-Lemaître-Robertson-Walker metric depends only on time. Differentiating the printed affine Lagrangian gives the canonical momentumThe Euler-Lagrange equations give $d{P}_i^c/d\tau =\partial L/\partial x^i=0$, soThis is the conserved quantity associated with spatial translation invariance. The overall minus sign in the stated action makes this canonical covector opposite to the usual future-directed mechanical momentum; it does not alter the geodesic trajectories or the conservation result. The canonical momentum sign in the affine particle action will be kept explicit below.