Symmetry structures and conserved forms of di erential equations on curved manifolds

Abstract

We explore two methods in nding Noether symmetries and conservation laws of di erential equations on Riemannian manifolds. The rst one is based on the Noether's theorem while the second one is about the `multiplier approach'. Using the rst method, we try to nd the variational symmetries, here, denoted X. With geodesic equations, the second method consists of nding the Lagrangian multipliers. This yields the conserved quantities when one acts the multipliers on the geodesic equations. It turns out that the total number of conserved quantities is equal to the number of variational symmetries found. The Lie algebra of in nitesimal isometries of the Riemannian manifolds studied has the dimension not exceeding 1 2n(n + 1), where n = 4. In the rst case studied in Chapter 2, variational symmetries and conservation laws of a modi ed de Sitter metric are classi ed. We came up with the suggestion that where a nite dimensional group generated by conservation laws exists, the Noether symmetry group has at least one additional symmetry that is not given by the Killing Vectors. This is later con rmed in the rest of all the other cases studied.