Separable and partially separable systems in the light of the inverse problem of dynamics

Motion On the Sphere: Integrability and Families of Orbits

“…As a special case, they also found two cases of monoparametric families of straight lines, whose presence guarantees the existence of a second integral, linear in the momenta. For analogous results in the case of three-dimensional motion, see Puel (1995) and for the case of the detection of multiseparability of the potential through the existence of certain biparametric families of orbits, see Voyatzi & Ichtiaroglou (2003). Bozis and Mertens (1985) presented the corresponding equations of the inverse problem for the case of a unit mass moving on an arbitrary surface S, under the influence of an unknown potential.…”

A Generalization of Szebehely's Inverse Problem of Dynamics

“…As before, since the right-hand side vanishes under the action of Z 0 , acting with Z 0 on the left will produce exactly the second-order pde for η which follows from (29).…”

“…The link between this equation and the second-order equation (29) for η also works in the same way as in the previous section. It suffices to introduce 2(E − V )/g(Z 0 , Z 0 ), which by the energy relation is simply our function η = h 2 , as new dependent variable in (38).…”

“…We can then right away impose the inverse problem condition (29), which reads g 12 η xx = g 11 η xy and integrates to g 11 η y − g 12 η x = φ(y), φ arbitrary. This is in fact the equation (41) for this case and it further integrates to η = σ(y) + ψ(g 11 x + g 12 y), with σ and ψ arbitrary functions of the indicated arguments.…”

“…With this expression for η, the relations (18,19) provide admissible forces which, in accordance with our choice for the multiplier g, should satisfy the relation (cf. (29)) bY x − aX y = 0. It is straightforward to verify that this is indeed the case and it implies that we should have (in agreement with (30,31)):…”