  In this thesis, we study the one parameter point transformations which leave
invariant the differential equations. In particular we study the Lie and the
Noether point symmetries of second order differential equations. We establish a
new geometric method which relates the point symmetries of the differential
equations with the collineations of the underlying manifold where the motion
occurs. This geometric method is applied in order the two and three dimensional
Newtonian dynamical systems to be classified in relation to the point
symmetries; to generalize the Newtonian Kepler-Ermakov system in Riemannian
spaces; to study the symmetries between classical and quantum systems and to
investigate the geometric origin of the Type II hidden symmetries for the
homogeneous heat equation and for the Laplace equation in Riemannian spaces. At
last but not least, we apply this geometric approach in order to determine the
dark energy models by use the Noether symmetries as a geometric criterion in
modified theories of gravity.
