Comprehensive analysis of the symmetries and conservation laws of the geodesic equations for a particular string inspired FRLW solution

• This paper is dedicated to a thorough investigation of the symmetries and conservation laws of the geodesic equations associated to a specific exact cosmological solution of a scalar-field potential which was originally motivated by six-dimensional Einstein-Maxwell theory.
• The Christoffel symbols and the corresponding system of geodesic equations are computed and then the associated Lie symmetries are totally analyzed.
• The algebraic structure of the Lie algebra of local symmetries is briefly investigated and a complete classification of the symmetry subalgebras is presented. Moreover, the corresponding invariant solutions of the system of geodesic equations are discussed.
• The Noether symmetries and the Killing vector fields of the geodesic Lagrangian are determined and the corresponding optimal system of one-dimensional subalgebras is constructed.
• An entire set of local conservation laws is computed for our analyzed scalar-field cosmological solution.

Scalar-field cosmology can be regarded as one of the significant fields of research in recent years. This paper is dedicated to a thorough investigation of the symmetries and conservation laws of the geodesic equations associated to a specific exact cosmological solution of a scalar-field potential which was originally motivated by six-dimensional Einstein-Maxwell theory. The mentioned string inspired Friedmann-Robertson-Lamai^tre-Walker (FRLW) solution is one of the noteworthy solutions of Einstein field equations. For this purpose, first of all the Christoffel symbols and the corresponding system of geodesic equations are computed and then the associated Lie symmetries are totally analyzed. Moreover, the algebraic structure of the Lie algebra of local symmetries is briefly investigated and a complete classification of the symmetry subalgebras is presented. Besides by applying the resulted symmetry operators the invariant solutions of the system of geodesic equations are discussed. In addition, the Noether symmetries and the Killing vector fields of the geodesic Lagrangian are determined and the corresponding optimal system of one-dimensional subalgebras is constructed. Mainly, an entire set of local conservation laws is computed for our analyzed scalar-field cosmological solution. For this purpose, two distinct procedures are applied: the celebrated Noether’s theorem and the direct method which is fundamentally based on a systematic application of Euler differential operators which annihilate any divergence expression identically.