GEOMETRICALLY SIMILAR ORBITS IN HOMOGENEOUS POTENTIALS

If V(r, theta) = r(m)G(theta) is, in polar coordinates, a homogeneous potential, of degree m, which can give rise to a given family f(r, the ta) = rg(theta) = constant of geometrically similar planar orbits, a s econd-order ordinary, linear in G(theta), homogeneous differential equ ation is found for any function g(theta) specifying the family. In con trast with the partial differential equation relating potentials with families, this ordinary equation is much easier to handle. For certain choices of g(theta) it can be solved to completion, for any m. Exampl es are offered. Two special cases refering to central potentials and i soenergetic families are studied.