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Normalization and the detection of integrability: The generalized Van Der Waals potential

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Abstract

Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.

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Author notes

  1. David Farrelly

    Present address: Department of Chemistry and Biochemistry, Utah State University, 84322-0300, Logan, UT, U.S.A.

Authors and Affiliations

  1. Harvard-Smithsonian Center for Astrophysics, Institute for Theoretical Atomic and Molecular Physics, 60 Garden Street, 02138, Cambridge, MA, U.S.A.

    David Farrelly

  2. School of Physics, Georgia Institute of Technology, 30332-0430, Atlanta, GA, U.S.A

    T. Uzer

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  1. David Farrelly
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  2. T. Uzer
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Farrelly, D., Uzer, T. Normalization and the detection of integrability: The generalized Van Der Waals potential. Celestial Mech Dyn Astr 61, 71–95 (1995). https://doi.org/10.1007/BF00051689

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  • DOI: https://doi.org/10.1007/BF00051689

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