A new formulation of the two-dimensional inverse problem of dynamics

Taking the parameter b to vary along a monoparametric family of planar curves, given in the form x = x(lambda, b), y = y(lambda, b) (lambda being the parameter along each specific curve of the family), we derive two equations to formulate the inverse problem of dynamics and find all potentials creating, for adequate initial conditions, the given family. One of these equations offers the total energy on each specific orbit traced under a known potential, the other equation relates merely potentials and orbital data. This later equation lends itself to series expansion solutions for small values of the parameter b. Two applications to isotach and geometrically similar orbits are discussed as special cases and two examples are given to demonstrate the efficiency and the indispensability of the new equations.