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Normal Forms for Hamiltonian Systems in Some Nilpotent Cases

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Abstract

We study Hamiltonian systems with two degrees of freedom near an equilibrium point, when the linearized system is not semisimple. The invariants of the adjoint linear system determine the normal form of the full Hamiltonian system. For work on stability or bifurcation the problem is typically reduced to a semisimple (diagonalizable) case. Here we study the nilpotent cases directly by looking at the Poisson algebra generated by the polynomials of the linear system and its adjoint.

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Notes

  1. It would also be possible to use \(\mathbf{H}=a(xX+yY)+\delta xY\) with \(a>0\) and \(\delta=\pm 1\), so that only positive times have to be taken into account.

  2. Indeed, for a differential equation of the form \(\dot{z}=\nabla L\times\nabla H\) both \(L\) and \(H\) are integrals since \(\dot{L}=\nabla L\cdot\dot{z}=\nabla L\cdot\nabla H\times\nabla L=0\) (the triple product with two common terms is zero). Similarly for \(H\).

  3. Another possibility is to use \(\mathbb{H}=a(xY-Xy)+\frac{\delta}{2}(x^{2}+y^{2})\) with \(a>0,\delta=\pm 1\) so that only positive time has to be considered.

  4. These invariants can also be obtained by transforming the invariants of Section 3 by the transformation in Section 3.7.

References

  1. Arnold, V. I., On Matrices Depending on Parameters, Russian Math. Surveys, 1971, vol. 26, no. 2, pp. 29–43; see also: Uspekhi Mat. Nauk, 1971, vol. 26, no. 2, pp. 101-114.

    Article  MathSciNet  Google Scholar 

  2. Birkhoff, G. D., Dynamical Systems, Providence, R.I.: AMS, 1966.

    MATH  Google Scholar 

  3. Deprit, A., Canonical Transformations Depending on a Small Parameter, Celestial Mech., 1969/70, vol. 1, no. 1, pp. 12–30.

    Article  MathSciNet  MATH  Google Scholar 

  4. Elphick, C., Tirapegui, E., Brachet, M. E., Coullet, P., and Iooss, G., A Simple Global Characterization for Normal Forms of Singular Vector Fields, Phys. D, 1987, vol. 29, no. 1–2, pp. 95–127.

    Article  MathSciNet  MATH  Google Scholar 

  5. Galin, D. M., Versal Deformations of Linear Hamiltonian Systems, in Sixteen Papers on Differential Equations, AMS Transl. (2), vol. 118, Providence, R.I.: AMS, 1982, pp. 1–12.

    Google Scholar 

  6. Laub, A. J. and Meyer, K., Canonical Forms for Symplectic and Hamiltonian Matrices, Celestial Mech., 1974, vol. 9, no. 2, pp. 213–238.

    Article  MathSciNet  MATH  Google Scholar 

  7. Lerman, L. M. and Markova, A. P., On Stability at the Hamiltonian Hopf Bifurcation, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 148–162.

    Article  MathSciNet  MATH  Google Scholar 

  8. Kummer, M., On Resonant Non Linearly Coupled Oscillators with Two Equal Frequencies, Comm. Math. Phys., 1976, vol. 48, no. 1, pp. 53–79.

    Article  MathSciNet  MATH  Google Scholar 

  9. Kummer, M., On Resonant Classical Hamiltonians with Two Equal Frequencies, Comm. Math. Phys., 1978, vol. 58, no. 1, pp. 85–112.

    Article  MathSciNet  MATH  Google Scholar 

  10. Meyer, K. R., Lie Transform Tutorial: 2, in Computer Aided Proofs in Analysis, K. R. Meyer, D. S. Schmidt (Eds.), IMA Vol. Math. Appl., vol. 28, New York: Springer, 1991, pp. 190–210.

    Chapter  Google Scholar 

  11. Meyer, K. R. and Schmidt, D. S., Periodic Orbits near \(\mathcal{L}_{4}\) for Mass Ratios near the Critical Mass Ratio of Routh, Celestial Mech., 1971, vol. 4, pp. 99–109.

    Article  MathSciNet  MATH  Google Scholar 

  12. Meyer, K. R. and Offin, D. C., Introduction to Hamiltonian Dynamical Systems and the \(N\)-Body Problem, 3rd ed., Appl. Math. Sci., vol. 90, Cham: Springer, 2017.

    Book  MATH  Google Scholar 

  13. Milnor, J., Morse Theory, Ann. Math. Stud., No. 51, Princeton, N.J.: Princeton Univ. Press, 1963.

    Book  MATH  Google Scholar 

  14. Palacián, J. and Yanguas, P., Reduction of Polynomial Planar Hamiltonians with Quadratic Unperturbed Part, SIAM Rev., 2000, vol. 42, no. 4, pp. 671–691.

    Article  MathSciNet  MATH  Google Scholar 

  15. Poincaré, H., Les méthodes nouvelles de la mécanique céleste: Vol. 3. Invariants intégraux. Solutions périodiques du deuxième genre. Solutions doublement asymptotiques, Paris: Gauthier-Villars, 1899.

    Book  MATH  Google Scholar 

  16. Schmidt, D., Versal Normal Form of the Hamiltonian Function of the Restricted Problem of Three Bodies near \(\mathcal{L}_{4}\), J. Comput. Appl. Math., 1994, vol. 52, no. 1–3, pp. 155–176.

    Article  MathSciNet  MATH  Google Scholar 

  17. Sokol’skii, A. G., Proof of the Stability of Lagrange Solutions at a Critical Relation of Masses, Sov. Astron. Lett., 1978, vol. 4, no. 2, pp. 79–81; see also: Pis’ma v Astron. Zh., 1978, vol. 4, no. 3, pp. 148-152.

    Google Scholar 

  18. Williamson, J., On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems, Amer. J. Math., 1936, vol. 58, no. 1, pp. 141–163.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, 45221-0025, Cincinnati Ohio, USA

    Kenneth R. Meyer

  2. Department of Electrical Engineering and Computer Science, University of Cincinnati, 45221-0030, Cincinnati Ohio, USA

    Dieter S. Schmidt

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  1. Kenneth R. Meyer
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  2. Dieter S. Schmidt
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Correspondence to Kenneth R. Meyer or Dieter S. Schmidt.

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MSC2010

34C40, 37C80, 37J15, 70H85

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Meyer, K.R., Schmidt, D.S. Normal Forms for Hamiltonian Systems in Some Nilpotent Cases. Regul. Chaot. Dyn. 27, 538–560 (2022). https://doi.org/10.1134/S1560354722050033

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