Programmed motion in the presence of homogeneity

Abstract

In the framework of the inverse problem of dynamics, we face the following question with reference to the motion of one material point: Given a region T orb of the xy plane, described by the inequality g(x, y) ≤ c0, are there potentials V = V (x, y) which can produce monoparametric families of orbits f (x, y) = c (also to be found) lying exclusively in the region T orb ? As the relevant PDEs are nonlinear, an answer to this question (generally affirmative, but not with assurance) can be given by the procedure of the determination of certain constants specifying the pertinent functions. In this paper we ease the mathematics involved by making certain simplifying assumptions referring to the homogeneity of both the function g(x, y) (describing the boundary of T orb ) and of the slope function γ(x, y) = fy/fx (representing the required family f (x, y) = c). We develop the method to treat the so formulated problem and we show that, even under these restrictive assumptions, an affirmative answer is guaranteed provided that two algebraic equations have in common at least one solution.