, Volume 21, Issue 4, pp 307-315

Application of Symmetries to Central Force Problems

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Abstract

We determine the Noether point symmetries associated with theusual Lagrangian of the differential equation of the orbit of the twobody problem. This gives rise to three definite natural forms (apartfrom the linear form) when the Lagrangian admits three Noether pointsymmetries which enables the solution of the orbit equation in terms ofelementary functions. The other forms for which the orbit equation hastwo point symmetries that arise in the Lie classification do not occurfor the usual Lagrangian. For central force problems we obtain inaddition to the six general central forces found by Broucke [8], newforce laws which lead to integrability in terms of known functions. Thisis achieved by the two Noether cases and further cases by use ofequivalence transformations of the orbit differential equation.Moreover, we give an extension of what is sometimes referred to asNewton's theorem of revolving orbits. We also make use of the Lieclassification to provide two new cases of integrable (in terms of knownfunctions) orbit differential equations and new force laws.