Equation (15) for the perturbation parameter ν is incorrect,1 and Eq. (14) associated with it should be deleted. For the case of an inverted cone with an opening angle θ, the correct equation is ν=3/(1+ξ2), where ξ=dz/dr=cotθ. For elliptical motion, ν = 1, which implies that ξ=2 and θ = 35.3°. The trajectory of a ball rolling inside an inverted cone with this opening angle is shown in Fig. 5. For a funnel with a hyperbolic profile, z = −k/r, where k is a constant, and the transverse force is an inverse square force proportional to dz/dr = k/r2, provided k/r2 ≪ 1. For a small eccentricity ϵ, ν = 1, and the transverse projection of the orbits is ellipses with one focus on the axis of the funnel. The trajectory of a ball rolling inside such a funnel is shown in Fig. 6. For a funnel with a parabolic profile, z = kr2 and ν = 2, and the orbits are approximate ellipses with the center along the axis of the cone.

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Perturbation approximation for orbits in axially symmetric funnels
Am. J. Phys.