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We show that the conservation laws for the geodesic equation which areassociated to affine symmetries can be obtained from symmetries of theLagrangian for affinely parametrized geodesics according to Noether's theorem,in contrast to claims found in the literature. In particular, using Aminova'sclassification of affine motions of Lorentzian manifolds, we show in detail howaffine motions define generalized symmetries of the geodesic Lagrangian. Wecompute all infinitesimal proper affine symmetries and the correspondinggeodesic conservation laws for all homogeneous solutions to the Einstein fieldequations in four spacetime dimensions with each of the followingenergy-momentum contents: vacuum, cosmological constant, perfect fluid, pureradiation, and homogeneous electromagnetic fields.

Affine Symmetry, Geodesics, and Homogeneous Spacetimes