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

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Abstract

In this present communication, the integrable modulation problem has been applied to study parametric extension of the Kapitza rotating shaft problem, which is a prototypical example of curl force as formulated by Berry and Shukla in (JPA 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 nonparametric cases.

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Acknowledgements

PG is deeply indebted to express his sincere gratitude and grateful thanks to Sir Professor Michael Berry for his enlightening discussions regarding the Airy function solutions to the Mathieu equations and also for his constant encouragements toward this work. PG & SG are also grateful to Professors Jayanta Bhattacharjee, Michele Bartuccelli, Guido Gentile, Anindya Ghose-Choudhury, Sumanto Chanda, and Pragya Shukla for their interests and valuable discussions. PG thanks Khalifa University of Science and Technology for its continued support toward this research work under the grant number FSU-2021-014. SG thanks Diamond Harbour Women’s University for providing the necessary research environment with constant encouragement and support.

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Authors and Affiliations

  1. Department of Mathematics, Khalifa University of Science and Technology, P.O. Box-127788, Abu Dhabi, UAE

    Partha Guha

  2. Department of Physics, Diamond Harbour Women’s University, D. H. Road, Sarisha, West Bengal, 743368, India

    Sudip Garai

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  1. Partha Guha
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  2. Sudip Garai
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Correspondence to Sudip Garai.

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Dedicated to Sir Michael Berry on his 80-th birthday with great respect and admiration.

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Guha, P., Garai, S. Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping. Nonlinear Dyn 106, 3091–3100 (2021). https://doi.org/10.1007/s11071-021-06947-6

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