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

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