Determination of autonomous three-dimensional force fields from a two-parameter family of orbits

Any two of the componentsX, Y, andZ of an autonomous force field which gives rise to the space orbitsF(x, y, z)=c ₁,G(x, y, z)=c ₂ are related by a partial differential equation with coefficients depending on the functionsF andG. This is a generalization of the corresponding equation for planar orbits (Bozis, 1983). The above partial differential equation is accompanied by the algebraic linear equation inX, Y, andZ expressing the fact that the force vector is lying in the osculating plane at each point of the orbit. The two equations constitute a generalization of the corresponding Szebehely's equations in the three dimensional space (Érdi, 1982). The generalization is meant in the sense that the dynamical system is not necessarily assumed to be conservative.