Application of symmetries to central force problems

Abstract

We determine the Noether point symmetries associated with the usual Lagrangian of the differential equation of the orbit of the two body problem. This gives rise to three definite natural forms (apart from the linear form) when the Lagrangian admits three Noether point symmetries which enables the solution of the orbit equation in terms of elementary functions. The other forms for which the orbit equation has two point symmetries that arise in the Lie classification do not occur for the usual Lagrangian. For central force problems we obtain in addition to the six general central forces found by Broucke, new force laws which lead to integrability in terms of known functions. This is achieved by the two Noether cases and further cases by use of equivalence transformations of the orbit differential equation. Moreover, we give an extension of what is sometimes referred to as Newton's theorem of revolving orbits. We also make use of the Lie classification to provide two new cases of integrable (in terms of known functions) orbit differential equations and new force laws.