Application of symmetries to central force problems (Q1566996)

The paper presents symmetry analysis of a single second-order differential equation which describes the orbits of central force problem. Work on the classification of forces for which an explicit solution can be obtained (in terms of circular functions and elliptic integrals) was presented in the standard book of \textit{E. T. Whittaker} [see e.g. A treatise on the analytical dynamics of particles and rigid bodies. With an introduction to the problem of three bodies. With a foreword by Sir William McCrea. Reprint of the fourth edition. Cambridge Mathematical Library, Cambridge (UK) etc.: Cambridge University Press. xvii (1988; Zbl 0665.70002)], and was later extended by \textit{R. Broucke} [Int. J. Eng. Sci. 17, 1151-1162 (1979; Zbl 0418.70025)]. In the present note, a rather peculiar mixture of arguments is used in an attempt to arrive at new cases of such integrable orbit equations: from Noether's theorem, where existence of a second symmetry is required next to the trivial translations of the independent variable, over the condition that these two symmetries constitute a two-dimensional Lie algebra, to an appeal to Lie's classification of such subalgebras. Most of the results are not really new, as they originate from known cases via a point transformation.