An alternative point of view on the equations of the inverse problem of dynamics

The version of the inverse problem of dynamics considered here is: given a family of planar curves f (x, y) = c, find the potentials V(x, y) which give rise to this family. Its solution is based on two linear partial differential equations satisfied by V: one of first order, containing the total energy function E(f), given by Szebehely in 1974, and the other one of second order, derived by Bozis in 1984 by eliminating the energy from Szebehely's equation. In this paper, Bozis' partial differential equation is obtained directly by eliminating the time derivatives of x (t) and y (t) up to the third order between seven differential relations based on the equations of motion and on the given family. Szebehely's equation is then derived as a consequence. This shows the importance of Bozis' equation, which is traditionally considered as following from Szebehely's one. The connection with the nonconservative case is emphasized.