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Crossing a nonlinear resonance

Adiabatic invariants and the Melnikov-Arnold integral

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Abstract

The idea of adiabatic invariance is presented in the context of simple classical mechanical models. The adiabatic invariant jumps across the separatrix — an attempt has been made to bring out the basic ideas underlying the Melnikov-Arnold integral. This becomes important as soon as a perturbation to a regular, stable system makes it dynamically unstable.

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Suggested Reading

  1. V I Arnold, Geometric theory of ordinary differential equations, Springer, 1983.

    Book  Google Scholar 

  2. E Fermi, Phys. Rev., Vol.75, p.1169, 1949.

    Article  Google Scholar 

  3. F S Crawford, Am. J. Phys., Vol.58, p.337, 1990.

    Article  Google Scholar 

  4. R Kulsrud, Phys. Rev., Vol.106, p.205, 1957.

    Article  Google Scholar 

  5. L D Landau, E M Lifshitz, Mechanics, Pergamon, 1969.

    Google Scholar 

  6. A B Rechester, T H Stix, Phys. Rev., A Vol.19, p.1656, 1979.

    Article  Google Scholar 

  7. J D Crawford, J Lott and H Riecke, Am. J. Phys., Vol.50, p.363, 1982.

    Article  Google Scholar 

  8. V I Arnold, V V Kozlov and A I Neishtadt, Mathematical aspects of classical and celestial mechanics, Third Ed., Springer, 2006.

    Google Scholar 

  9. A J Lichtenberg and M A Lieberman, Regular and chaotic dynamics, Second Ed., Springer, 1992.

    Book  Google Scholar 

  10. V I Arnold, Sov. Phys. Dokl., Vol.161, p.1, 1965.

    Google Scholar 

  11. B Chirikov, Phys. Rep., Vol.52, p.263, Appendix, 1979.

    Article  Google Scholar 

  12. P J Holmes, Phys. Rep., Vol.193, p.137, 1990.

    Article  Google Scholar 

  13. P J Holmes and J E Marsden, J. Math. Phys., Vol.23, p.669, 1982.

    Article  Google Scholar 

  14. G M Zaslavsky, Physics of chaos in Hamiltonian systems, Second Ed., Imperial College Press, 2007.

    Book  Google Scholar 

  15. S M Soskin, P V E McClintock, T M Fromhold, I E Khovanov and R Manella, Contemporary Phys., Vol.51, p.233, 2010.

    Article  Google Scholar 

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Authors and Affiliations

  1. Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai, 400085, India

    Sudhir R. Jain

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  1. Sudhir R. Jain
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Correspondence to Sudhir R. Jain.

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Sudhir R Jain is working at the Nuclear Physics Division, Bhabha Atomic Research Centre. He is associated with the faculty of physical sciences, Homi Bhabha National Institute, and is an adjunct Professor at the UM-DAE Centre for Excellence in Basic Sciences. His main interests are quantum chaos, connections of chaos with statistical mechanics, nuclear theory, and mathematical physics.

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Jain, S.R. Crossing a nonlinear resonance. Reson 19, 797–813 (2014). https://doi.org/10.1007/s12045-014-0089-8

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  • DOI: https://doi.org/10.1007/s12045-014-0089-8

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