Curvature in orbital dynamics

The physical basis and the geometrical significance of the equation for the orbit of a particle moving under the action of external forces is exhibited by deriving this equation in a coordinate-independent representation in terms of the radius of curvature of the orbit. Although this formulation appeared in Newton's Principia, it has been ignored in contemporary classical mechanics textbooks. For small eccentricities, the orbit equation is used to obtain approximate solutions that illustrate the role of curvature. It is shown that this approach leads to a simple graphical method for determining the orbits for central forces. This method is similar to one attributed to Newton, who applied it to a constant central force, and sent a diagram of the orbit to Hooke in 1679. The result is compared to the corresponding orbit of a ball revolving inside an inverted cone which Hooke described in his response to Newton.