On monoparametric families of orbits sufficient for integrability of planar potentials with linear or quadratic invariants

Motion On the Sphere: Integrability and Families of Orbits

“…By noting the similarity of Bozis' equation to the equation of Darboux and identifying the two equations, Ichtiaroglou and Meletlidou (1990) found all the cases of monoparametric families of orbits, for which the two equations coincide, i.e., they found the monoparametric families of orbits whose presence guarantees the existence of a second integral, quadratic in the momenta and, thus, the integrability of V (x, y). 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.…”

Motion on a given surface: Monoparametric families of orbits sufficient for separability

“…From the case of planar motion, where the existence of suitable families of conics guarantees also separability [10] we know that this condition is in general not also necessary. It would be interesting to search for the existence of other permissible monoparametric families of orbits with this property, for the case of motion on a two-dimensional surface in a general setting.…”

“…Then we have f u = 1, f v = 0 and taking into account (10) and (11), the quantities α and β in (8) are written as:…”

“…We consider motion on R 2 and assign the metric These results are already known as special cases in Ichtiaroglou and Meletlidou [10]. The compatibility of the above potential with the family of circles was initially given in Broucke and Lass [6].…”

“…Ichtiaroglou and Meletlidou [10] have shown that, in the case of planar motion, the presence of a monoparametric family of conic sections or a family of confocal parabolas guarantees the integrability of the potential, with an integral of motion quadratic in the velocities. As special cases, they obtained that the presence of a family of straight lines, intersecting at a certain point, results to a central potential and a linear integral of motion (i.e.…”