r

in an axisymmetric gravitational potential,

Ω^{2} = \frac{1}{r} \frac{\partial Φ}{\partial r} (r, 0) .

A small vertical displacement satisfies

\overset{z}{¨} = - \frac{\partial ^{2} Φ}{\partial z ^{2}} (r, 0) z,

Ω_{z}^{2} = \frac{\partial ^{2} Φ}{\partial z ^{2}} (r, 0) .

Φ (R)

, where

R = (r^{2} + z^{2})^{1/2}

\frac{\partial ^{2} Φ}{\partial z ^{2}} (r, 0) = \frac{1}{r} \frac{d Φ}{d R} (r) = Ω^{2} .

Hence

Ω_{z} = Ω

. Geometrically, a slightly tilted circular orbit remains a circular orbit in a different plane, and its height completes one oscillation per revolution.

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