General Relativity and Gravitation

, Volume 42, Issue 12, pp 2957–2980 | Cite as

Lie and Noether symmetries of geodesic equations and collineations

Research Article


The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.


Geodesics General relativity Classical mechanics Collineations Lie symmetries Projective collineations Noether symmetries First integrals 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Physics, Department of Astronomy-Astrophysics-MechanicsUniversity of AthensAthensGreece

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