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Finding potentials or force fields from given families of orbits is the type of inverse problem of dynamics discussed in this paper. We present the pertinent partial differential equations for various versions of the problem such as, for instance, for conservative or autonomous non-conservative fields in two or three dimensions, for inertial or relating frames, for one material point or, more, generally, for holonomic systems with n degrees of freedom. The notion of the family boundary curves is introduced. The role of the homogeneity of the given family or of the required potential, as well as the question of multitude of compatible pairs of orbits and potentials is discussed. Comments on the relation of the problem to problems of astronomical interest are made, at appropriate places, throughout the text.

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We study three-dimensional homogeneous potentials V (x, y, z) = x m R y x , z x of degree m from the viewpoint of their compatibility with preassigned two-parametric families of spatial regular orbits given in the solved form f (x, y, z) = c 1 , g(x, y, z) = c 2 where each of the functions f and g is also homogeneous in x, y, z (of any degree n f and n g , respectively) and where c 1 , c 2 are real constants. The orbital elements identifying each family may be represented uniquely by a pair of functions α(x, y, z) and β(x, y, z), both homogeneous of zero degree. Then, depending on the case at hand, we find certain differential conditions (including m and partial derivatives of the given orbital elements) which, if satisfied, ensure the existence of a homogeneous potential V (x, y, z) generating the preassigned orbits. Finally, we offer certain examples which cover the general case as well as special cases.