Inverse problem of dynamics, Galiullin and Szebehely methods and curl force trajectories

At first we study Galiullin's construction of Bertrand problem and compare it with Szebehely's method, latter is based on a first order partial differential equation for the unknown potential that produces a prescribed monoparametric family of planar trajectories. In the second part of the paper we study the inverse problem of the trajectories such that the corresponding force is a nonconservative position dependent one, satisfying the non-vanishing curl condition and not the gradient of a potential function. Recently this force has been introduced and popularized by Berry and Shukla (J. Phys. A 45 (2012) 305 201). We connect the inverse problem dynamics of these curl force trajectories with the generalized potentials obtained by Sarlet-Mestdag-Prince (Rep. Math. Phys. 72(2013) 65-84) from the inverse problem of phi(x, y) = xy m for m not equal 0, m not equal -1. Finally we show that the analog of these curly trajectories in momentum space can be manifested as kinetic energies of the pair of Calogero-Leyvraz Hamiltonians (J. Nonlinear Math. Phys. 26 (2019) 147-154) describing the motion of a particle in a magnetic field with friction.