Parametric resonance and geodesics on an ellipsoid (Q1324677)

Let a material point of mass \(m\) move on the line in the field with potential \({k(t) \over 2} x^ 2\), where \(k(t)\) is a positive periodic function of time. The main result gives the answer to the question: for which \(k(t)\), the equilibrium in this system is stable for all sufficiently small \(m\). The answer turns out to be closely related to the Jacobi problem about geodesics on an ellipsoid.