### Abstract:

The only central force laws which result in closed stable orbits for. all bounded motion (for a range of values of nonzero measure of the energy E and the angular momentum L) are the Kepler problem (inverse square law) and Hooke's law (isotropic harmonic oscillator). This result was first proved by J Bertrand and is referred to as Bertrand's theorem. An application of Lie group theory to the differential equation of the orbit for arbitrary central force laws helps us tb make a connection between the symmetry properties of the differential equation of the orbit and the conditions under which Bertrand's theorem hold. The group classification of the differential equation of the orbit for arbitrary central force laws is carried out with respect to exact Lie point symmetries and Noether point symmetries. <br><br> We then investigate the existence of closed bound orbits in a plane and derive the necessary and sufficient conditions for the most general equation of motion in a plane with variable angular momentum to represent closed elliptical orbits. Furthermore, using the theory of approximate symmetry groups, we extend its application to our study of the group properties of the differential equation of the orbit for the most general equation of motion in a plane that gives rise to closed orbits. <br><br> In the analysis of differential equations the relationship between first integrals and symmetries is of extreme importance. Due to the direct relevance of this relationship to the present work, we provide a complete classification of first integrals of scalar second-order ordinary differential equations according to the real Lie algebra of point symmetries they admit. We show that the real maximal Lie algebra of the first integrals of any scalar second-order ordinary differential equation is a proper subalgebra of the equation itself. Moreover, we provide a connection between integrability of scalar second-order ordinary differential equations and the existence of no point symmetry.