Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping

In this present communication the integrable modulation problem has been applied to study parametric extension of the Kapitza rotating shaft problem, which is a protypical example of curl force as formulated by Berry and Shukla in [J. Phys. A 45 305201 (2012)] associated with simple saddle potential. The integrable modulation problems yield parametric time dependent integrable systems. The Hamiltonian and first integrals of the linear and nonlinear parametric Kapitza equation (PKE) associated with simple and monkey saddle potentials have been given. The construction has been illustrated by choosing $\omega(t) = a + b \cos t$ and that maps to Mathieu type equations, which yield Mathieu extension of PKE. We study the dynamics of these equations. The most interesting finding is the mixed mode of particle trapping and escaping via the heteroclinic orbits depicted with the parametric Mathieu-Kapitza equation which are absent in the case of non parametric cases.