A Survey of Normalization Techniques Applied to Perturbed Keplerian Systems

  • Richard H. Cushman
Part of the Dynamics Reported book series (DYNAMICS, volume 1)

Abstract

This article is a survey of the use of normal form techniques in the study of Hamiltonian perturbations of the two body problem in three space. We treat the following examples in some detail: the quadratic Zeeman effect, orbit­ing dust, a three dimensional lunar problem, and the main problem of artificial satellite theory.

Keywords

Dust Manifold Weinstein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Marsden, J. and Weinstein, A., Reduction of manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121–130.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arms, J., Cushman, R.H., and Got ay, M.J., A universal reduction procedure for Hamiltonian group actions, (to appear in Proc. Integrable Hamiltonian Systems, eds. T. Ratiu and H. Flaschka, MSRI series, Springer Verlag, 1990).Google Scholar
  3. 3.
    Abraham, R. and Marsden, J., “Foundations of Mechanics”, Benjamin/Cummings, Reading, Mass., 1978.Google Scholar
  4. 4.
    Duistermaat, J.J., Bifurcations of periodic solutions near equilibrium points of Hamiltonian systems, in: “Bifurcation Theory and Applications”, Montecatini, 1983, 55–105, ed. L. Salvadori, Lect. Notes in Math., 1057, Springer Verlag, New York, 1984.Google Scholar
  5. 5.
    van derMeer, J.-C., The Hamiltonian Hopf bifurcation, Lect. Notes in Math., 1160, Springer Verlag, New York, 1985.MATHGoogle Scholar
  6. 6.
    Cotter, C., The semisimple 1:1 resonance, Thesis, University of California at Santa Cruz, 1986.Google Scholar
  7. 7.
    Arnold, V.l., “Mathematical Methods in Classical Mechanics”, Graduate Texts in Math., 60, Springer Verlag, New York, 1978Google Scholar
  8. 8.
    Coffey, S., Deprit, A., and Williams, C.A., The quadratic Zeeman effect, Ann. N.Y. Acad. Sci.,497 (1986), 22–36.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zimmerman, M.L., Kash, M.M., and Kleppner, D., Evidence of an approximate symmetry for hydrogen in a uniform magnetic field, Phys. Rev. Lett., 45 (1980), 1092–94.CrossRefGoogle Scholar
  10. 10.
    Solovev, E.A., The hydrogen atom in a weak magnetic field, Sov. Phys. JETP, 55(1982), 1017–1021.Google Scholar
  11. 11.
    Reinhardt, W. and Farelley, D., The quadratic Zeeman effect in hydrogen, J. de Physique, coll. 2, suppl. vol 43. no. 11 (1982), 29–43.Google Scholar
  12. 12.
    Deprit, A. and Ferrar, S., On polar orbits for the Zeeman effect in a moderately strong magnetic field, preprint, NIST, Gaithersburg, MD, 1989.Google Scholar
  13. 13.
    Cushman, R., Normal form for Hamiltonian vectorfields with periodic flow, in: “Differential Geometric Methods in Mathematical Physics”, 125–144, ed. S. Sternberg, Reidel, Dordrecht, 1984.Google Scholar
  14. 14.
    Cushman, R. and Sanders, J.A., The constrained normal form algorithm, Celest. Mech., 45 (1989), 181–187.MathSciNetMATHGoogle Scholar
  15. 15.
    Bertaux, J.L. and Blamont, J.E., Interpretation of Ogo 5 Lyman alpha measurements in the upper geocorona, J. Geophys. Res., 78 (1973), 80–91.CrossRefGoogle Scholar
  16. 16.
    van der Meer, J-C. and Cushman, R., Orbiting dust under radiation pressure, in: “Differential Geometric Methods in Theoretical Physics”, 403–414, ed. H. Doeb- ner, World Scientific, Singapore, 1987.Google Scholar
  17. 17.
    Conley, C., On some new long periodic solutions of the plane restricted three body problem, Comm. Pure Appl. Math., 41 (1963), 449–467.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kummer, M., On the stability of Hill’s solutions of the planar restricted three body problem, Amer. J. Math., 101 (1979), 1333–1345.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kummer, M., On the 3-dimensional lunar problem and other perturbations of the Kepler problem, J. Math. Anal. Appl., 93 (1983), 142–194.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    van der Meer, J-C. and Cushman, R., Constrained normalization of Hamiltonian systems and perturbed Keplerian motion, J. Appl. Math, and Phys. (ZAMP), 37 (1986), 402–424. MATHCrossRefGoogle Scholar
  21. 21.
    Brouwer, D., Solution of the problem of artificial satellite theory without drag, Astron. J., 64 (1959), 378–397.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Orlov, V.V., Almost circular periodic motions of a particle of matter under the gravitational attraction of a spheroid, Reports of the state astronomical institute in Shternberg, no. 88–89, Moscow University, 1953.Google Scholar
  23. 23.
    Eckstein, M., Shi, Y., and Kevorkian, J., Satellite motion for all inclinations around an oblate planet, in: IAU Symposium 25, 291–322, ed. G. Contopoulos, Academic Press, London, 1966.Google Scholar
  24. 24.
    Cushman, R., Reduction, Brouwer’s Hamiltonian and the critical inclination, Celest. Mech., 31 (1983), 409–429; errata Celest. Mech., 33 (1984), 297.MathSciNetGoogle Scholar
  25. 25.
    Coffey, S., Deprit, A., and Miller, B., The critical inclination in artificial satellite theory, Celest. Mech., 39 (1986), 365–405.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Cushman, R., An analysis of the critical inclination problem using singularity theory, Celest. Mech., 42 (1988), 39–51.MathSciNetGoogle Scholar
  27. 27.
    Moser, J., Regularization of Kepler’s problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 604–636.CrossRefGoogle Scholar
  28. 28.
    Milnor, J., On the geometry of the Kepler problem, Amer. Math. Monthly, 90 (1983), 353–365.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Weinstein, A., Poisson structures and Lie algebras, in: “The Mathematical Heritage of Élie Cartan”, Asterique (1985), Numéro Hors Sèrie, 421–434Google Scholar
  30. 30.
    Deprit, A., Canonical transformations depending on a parameter, Celest. Mech., 1 (1969), 12–30.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Baider, A., and Churchill, R., The Campbell Baker Hausdorff group, Pac. J. Math., 131 (1988), 219–235.MathSciNetMATHGoogle Scholar
  32. 32.
    Arnold, V., “Geometrical Methods in the Theory of Ordinary Differential Equations”, Gründl. Math. Wiss., vol. 260, Springer Verlag, New York, 1983MATHGoogle Scholar
  33. 33.
    Birkhoff, G.D., “Dynamical Systems”, AMS colloquium publications, vol. 9, AMS, Providence, 1927; revised ed. 1966.Google Scholar
  34. 34.
    Deift, P., Lund, F. and Trubowitz, E., Nonlinear wave equations and constrained harmonic motion, Comm. Math. Phys., 74 (1980), 144–188MathSciNetCrossRefGoogle Scholar
  35. 35.
    Moser, J., Geometry of quadrics and spectral theory, in: “The Chern Symposium 1979”, 147–188, ed. W.Y. Hsiang, et al. Springer Verlag, New York, 1980Google Scholar
  36. 36.
    Cushman, R. and Rod, D.L., Reduction of the 1:1 semisimple resonance, Physica D, 6 (1982), 105–112.MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Cushman, R. and Knörrer, H., The energy momentum mapping of the Lagrange top, in: “Differential Geometric Methods in Physics”, 12–24, ed. H. Doebner et al., Lecture notes in math., 1139, Springer Verlag, New York, 1985CrossRefGoogle Scholar
  38. 38.
    Steenrod, N., “The Topology of Fiber Bundles”, Princeton Univ. Press, Princeton, 1951.Google Scholar
  39. 39.
    Billera, L., Cushman, R. and Sanders, J., The Stanley decomposition of the harmonic oscillator, Proc. Ned. Akad. Wet., Series A 91 (1988), 375–393MathSciNetGoogle Scholar
  40. 40.
    Hirsch, M., “Differential Topology”, Graduate texts in math., vol. 33, Springer Verlag, New York, 1976Google Scholar
  41. 41.
    Smale, S., Diffeomorphisms of the 2-sphere, Proc. AMS, 10 (1959), 621–629.MathSciNetMATHGoogle Scholar
  42. 42.
    Smale, S., Topology and mechanics, Inventiones Math., 10 (1970), 305–331.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    van der Meer, J-C., On integrability and reduction of normalized perturbed Keplerian systems, preprint, Technical University of Eindhoven, 1988.Google Scholar
  44. 44.
    Mumford, D. “Algebraic Geometry I: Complex Projective Varieties”, Gründl. Math. Wiss. 221, Springer Verlag, New York, 1976. MATHGoogle Scholar
  45. 45.
    Arbarello, E., Cornalba, M., Griffiths, P.A., and Harris, J., “Geometry of Algebraic Curves I”, Springer Verlag, New York, 1985.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Richard H. Cushman
    • 1
  1. 1.Mathematics InstituteRijksuniversiteit UtrechtUtrechtThe Netherlands

Personalised recommendations