  We derive the Lie and the Noether conditions for the equations of motion of a
dynamical system in a $n-$dimensional Riemannian space. We solve these
conditions in the sense that we express the symmetry generating vectors in
terms of the special projective and the homothetic vectors of the space.
Therefore the Lie and the Noether symmetries for these equations are geometric
symmetries or, equivalently, the geometry of the space is modulating the motion
of dynamical systems in that space. We give two theorems which contain all the
necessary conditions which allow one to determine the Lie and the Noether
symmetries of a specific dynamical system in a given Riemannian space. We apply
the theorems to various interesting situations covering Newtonian 2d and 3d
systems as well as dynamical systems in cosmology.
