Programmed motion for a class of families of planar orbits

Taking as a guide the case of the set of monoparametric families y = h(x) + c, for which Szebehely's equation can be solved by quadratures for the potential V(x, y) generating the given set of orbits, we propose the following programmed motion problem: can we manage so as to have members of the given set inside a preassigned domain T subset of R-2 of the xy plane? We come to understand that, among the various inequalities by means of which T can be ascribed, the simplest is b(x, y) greater than or equal to 0 where, for each h(x), the function b(x, y) is related to the kinetic energy of the moving point (equations (19)-(21)). We then proceed to show that, in general, if b(x, y) satisfies two conditions (equations (39) and (40)), the answer to our question is affirmative: on the grounds of the given appropriate b(x, y), a function h(x) is found, associated with a certain potential V(x, y) creating members of the family y = h(x) + c inside the region b(x, y) greater than or equal to 0. Some special cases which stem from the method are studied separately. The limitations and also the promising features of the method developed to face the above inverse problem are discussed.