Skip to main content
SpringerLink
Log in

Perturbation Methods for Hamiltonian Systems. Generalizations

  • Chapter
Perturbation Methods in Non-Linear Systems

Part of the book series: Applied Mathematical Sciences ((AMS,volume 8))

  • 452 Accesses

Abstract

This chapter is devoted to two main goals. First introduce the reader to known methods of canonical perturbations, describe them in a heuristic way and give examples so as to motivate the theorems presented in Chapters III and IV. Second, present some basic results about iterative procedures of fundamental importance on methods of averaging. Major contributors to this area are Lindstedt (1884), Poincaré (1893), Whittaker (1916), Siegel (1941), Krylov (1947), Bogoliubov (1945), Kolmogorov (1953), Arnol’d (1963), Diliberto (1961), Pliss (1966), Kyner (1961), Moser (1962), Hale (1961) with several overlappings in results. Many of these results have been unified and consolidated in celebrated books by Siegel (1956), Wintner (1947), Newytskii-Stepanov (1960), Cesari (1963), Hale (1969), Abraham (1967), Birkhoff (1927), Bogoliubov-Mitropolskii (1961), Lefschetz (1959), Minorsky (1962), Sansone-Conti (1964), Sternberg (1970).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abraham, R., 1967, “Foundations of Mechanics”, W. A. Benjamin, Inc., Philadelphia.

    MATH  Google Scholar 

  2. Andoyer, H., 1926, “Cours de Mécanique Célèste”, (Vol. I), Gauthier-Villars, Paris.

    MATH  Google Scholar 

  3. Arnol’d, V. I., 1963, “Proof of A. N. Kolmogorov’s Theorem on the Conservation of Quasiperiodic Motions under Small Perturbations of the Hamiltonian”, Uspekhi Mat. Nauk USSR, 18, 13–40.

    Google Scholar 

  4. Arnol’d, V. I., 1963? “Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics”, Uspekhi Mat. Nauk USSR, 18, 91–192.

    Google Scholar 

  5. Barbanis, B., 1966, “The Topology of the Third Integral”, Intern. Astrom. Union Symp. No. 25, pp. 19–25, Academic Press, New York.

    Google Scholar 

  6. Birkhoff, G. D., 1927, “Dynamical Systems”, Am. Math. Soc. Colloq. Public. IX, Providence, Rhode Island.

    MATH  Google Scholar 

  7. Bogoliubov, N. N., 1945, “On some statistical methods in mathematical physics”, (Paper) Izv. Akad. Nauk USSR, Moscow.

    Google Scholar 

  8. Bogoliubov, N. N. and Mitropolskii, Y. A., 1951, “Asymptotic Methods in the Theory of Nonlinear Oscillations”, Gordon and Breach, New York.

    Google Scholar 

  9. Bozis, G., 1966, “A New Integral in the Restricted Problem of Three Bodies”, Doctoral Thesis, Univ. of Thessaloniki, Greece.

    Google Scholar 

  10. Bozis, G., 1966, “On the Existence of a New Integral in the Restricted Three-Body Problem”, Astron. J., 71, 404–414.

    Article  Google Scholar 

  11. Brouwer, D., 1959, “Solution of the Problem of Artificial Satellites without Drag”, Astron. J., 64, 378–390.

    Article  MathSciNet  Google Scholar 

  12. Brouwer, D. and Clemence, G. M., 1961, “Methods of Celestial Mechanics”, Academic Press, New York.

    Google Scholar 

  13. Caley, A., 1848, Comb. Dublin Math. J., 3, 116.

    Google Scholar 

  14. Cesari, L., 1940, “Sulla Stabilità delle Soluzioni dei Sistemi di Equazioni Differenziali Lineari a Coefficienti Periodici”, Atti Accad. Ital. Mem. Classe Fis. Mat. e Nat., 11, 633–692.

    MathSciNet  Google Scholar 

  15. Cesari, L., 1963, “Asymptotic Behavior and Stability Problems in Ordinary Differential Equations”, Springer-Verlag New York Inc., New York.

    MATH  Google Scholar 

  16. Choi, J. S. and Tapley, B. D., 1972, “An Extended Canonical Perturbation Method”, Cel. Mech. (to appear).

    Google Scholar 

  17. Contopoulos, G., 1960, “A Third Integral of Motion in a Galaxy”, Zeits. fur Astrophys., 49, 273–291.

    MATH  MathSciNet  Google Scholar 

  18. Contopoulos, G. and Barbanis, B., 1961, “An Application of the Third Integral of Motion”, The Observatory, 82, 80–82.

    Google Scholar 

  19. Contopoulos, G., 1962, “On the Existence of a Third Integral”, Astron. J., 68, 1–14.

    Article  MathSciNet  Google Scholar 

  20. Contopoulos, G., 1963, “A Classification of the Integrals of Motion”, Astrophys. J., 138, 1297–1305.

    Article  MathSciNet  Google Scholar 

  21. Contopoulos, G., 1963, “Resonances Cases and Small Divisors in a Third Integral of Motion. I”, Astron. J., 68,

    Google Scholar 

  22. Contopoulos, G. and Woltjer, L., 1964, “The Third Integral in Non-Smooth Potentials”, Astrophys. J., 140, 1106–1119.

    Article  MathSciNet  Google Scholar 

  23. Contopoulos, G., 1965, “The Third Integral in the Restricted Three-Body Problem”, Astrophys. J., 142, 802–804.

    Article  MATH  Google Scholar 

  24. Contopoulos, G., 1966, “Adiabatic Invariants and the Third Integral”, J. Math. Phys., 7, 788–797.

    Article  MathSciNet  Google Scholar 

  25. Contopoulos, G. and Hadjidemetrious, J. P., 1968, “Characteristics of Invariant Curves of Plane Orbits”, Astron. J., 73, 86–96.

    Article  Google Scholar 

  26. Contopoulos, G., 1970, “Resonance Phenomena in Spiral Galaxies”, in “Periodic Orbits, Stability and Resonances” (Ed. G. E. O. Giacaglia), D. Reidel Publ. Co., Dordrecht, Holland.

    Google Scholar 

  27. Contopoulos, G., “Orbits in Highly Perturbed Dynamical Systems”, I (1970), Astron. J., 75, 96–107;

    Article  MathSciNet  Google Scholar 

  28. II(1970), Astron. J., 75, 108–130;

    Article  MathSciNet  Google Scholar 

  29. III(1971), Astron. J., 76, 147–156.

    Article  MathSciNet  Google Scholar 

  30. Deprit, A. et. al., 1969, “Birkhoff’s Normalization”, Cel. Mech., 1, 222–251.

    Article  MATH  MathSciNet  Google Scholar 

  31. Deprit, A. et. al., 1970, “Analytical Lunar Ephermeris: Brouwer’s Suggestion”, Astron. J., 75, 747–750.

    Article  Google Scholar 

  32. Deprit, A. and Rom, A., 1970, “Characteristic Exponents of L4 in the Elliptic Restricted Problem”, Astron. Astrophys., 5, 416–428.

    MATH  MathSciNet  Google Scholar 

  33. Deprit, A. and Rom, A., 1970, “The Main Problem of Artificial Sattelite Theory for Small and Moderate Eccentricities”, Cel. Mech., 2, 166–206.

    Article  MATH  MathSciNet  Google Scholar 

  34. Diliberto, S. P., 1967, “New Results on Periodic Surfaces and the Averaging Principle”, U.S.-Japanese Sem. on Diff. Funct. Eq., pp. 49–87, Benjamin, Philadelphia.

    Google Scholar 

  35. Dirac, P. A. M., 1958, “Generalized Hamiltonian Dynamics”, Proc. Roy. Soc. London A246, 326–332.

    MathSciNet  Google Scholar 

  36. Euler, L., 1753, “Theoria Motus Lunae” and 1772, “Theoria Motus Lunae, Novo Methodo”, Petropoli, Typ. Acad. Imp. Scient.

    Google Scholar 

  37. Gelfand, I. M. and Lidskii, U. B., 1958, “On the Structure of the Regions of Stability of Linear Canonical Systems of Differential Equations with Periodic Coefficients”, Am. Math. Soc. Transi. (2), 8, 143–182.

    MathSciNet  Google Scholar 

  38. Giacaglia, G. E. O., 1964, “Notes on von Zeipel’s Method”, GSFC-NASA Publ. X-547-64-161.

    Google Scholar 

  39. Giacaglia, G. E. O., 1965, Evaluation of Methods of Integration by Series in Celestial Mechanics”, Doctoral Thesis, Yale University, New Haven.

    Google Scholar 

  40. Giacaglia, G. E. O., 1967, “Nonintegrable Dynamical Systems”, Chair Thesis, General Mechanics, Univ. of Sao Paulo, Sao Paulo.

    Google Scholar 

  41. Giacaglia, G. E. O., et al., 1970, “A Semi-Analytic Theory for the Motion of a Lunar Satellite”, Cel. Mech., 3, 3–66.

    Article  MATH  MathSciNet  Google Scholar 

  42. Giacaglia, G. E. O. and Jefferys, W. H., 1971, “Motion of a Space Station”, Cel. Mech. 4, 442–467.

    Article  MATH  Google Scholar 

  43. Giacaglia, G. E. O., 1971, “Characteristic Exponents at L4 and L5 in the Elliptic Restricted Problem of Three Bodies”, Cel. Mech., 4, 468–489.

    Article  MATH  MathSciNet  Google Scholar 

  44. Giacaglia, G. E. O., 1972, “Regularization of Conservative Central Fields”, Public. Astron. Soc. Japan, 24, No. 3, July.

    Google Scholar 

  45. Giacaglia, G. E. O. and Nuotio, V. I., 1972, “Spinor Regularization of Conservative Central Fields”, 3rd Annual Meeting, Div. Dynamical Astron., Am. Astron. Soc, Univ. of Maryland, In “Bull. Amer. Astron. Soc”, (to appear).

    Google Scholar 

  46. Goldstein, H., 1951, “Classical Mechanics”, Addison-Wesley, Reading, Massachusetts.

    MATH  Google Scholar 

  47. Goursat, E., 1959? “Cours d’Analyse Mathematique”, 7th. Ed., Vol. 2, Gauthier-Villars, Paris. (Reprinted by Dover Publ., New York).

    Google Scholar 

  48. Hale, J. K., 1954, “On the Boundedness of the Solution of Linear Differential Systems with Periodic Coefficients”, Riv. Mat. Univ. Parma, 5, 137–167.

    MATH  MathSciNet  Google Scholar 

  49. Hale, J. K., 1961, “Integral Manifolds of Perturbed Differential Equations”, Ann. Math., 73, 496–531.

    Article  MATH  MathSciNet  Google Scholar 

  50. Hale, J. K., 1962, “On Differential. Equations Containing a Small Parameter”, Contrib. Diff. Eq., Vol. 1, J. Wiley, New York (Ed. J. P. LaSalle et. al).

    Google Scholar 

  51. Hale, J. K., 1963, “Oscillations in Nonlinear Systems”, McGraw-Hill, New York.

    MATH  Google Scholar 

  52. Hale, J. K., 1969, “Ordinary Differential Equations”, (Chapt. 5), Wiley-Interscience, New York.

    MATH  Google Scholar 

  53. Henrand, J., 1970, “On a Perturbation Theory Using Lie Transforms”, Cel. Mech., 3, 107–120.

    Article  Google Scholar 

  54. Hènon, M. and Heiles, C., 1964, “The Applicability of the Third Integral of Motion; Some Numerical Experiments”, Astron. J., 69, 73–79.

    Article  Google Scholar 

  55. Hènon, M. and Heiles, C., 1945, “Exploration Numérique du Problème Restreint”, Ann. Astrophys., 28, 499

    Google Scholar 

  56. Hènon, M. and Heiles, C., 1945, “Exploration Numérique du Problème Restreint”, Ann. Astrophys., 28, 992.

    Google Scholar 

  57. Hori, G., 1966, “Theory of General Perturbations with Unspecified Canonical Variables”, Public. Astron. Soc. Japan, 18, 287–296.

    Google Scholar 

  58. Hori, G., 1971, “Theory of General Perturbations for Noncanonical Systems”, Publ. Astron. Soc. Japan, 23, 567–587.

    Google Scholar 

  59. Kamel, A. A., 1969? “Expansion Formulae in Canonical Transformations Depending on a Small Parameter”, Cel. Mech., 1, 190–199.

    Article  MATH  MathSciNet  Google Scholar 

  60. Kamel, A. A., 1970, “Perturbation Method in the Theory of Nonlinear Oscillations”, Cel. Mech., 3, 90–106.

    Article  MATH  MathSciNet  Google Scholar 

  61. Kevorkian, J., 1966, “The Two Variable Expansion Procedure for the Approximate Solution of Certain Nonlinear Differential Equations” in “Lectures in Applied Mathematics”, vol. 7, p. 206, Amer. Math. Soc, Providence, R. I.

    Google Scholar 

  62. Kolmogorov, A. N., 1953, “On the Conservation of Quasiperiodic Motions for a Small change in the Hamiltonian Function”, Dokl. Akad. Nauk USSR, 98, 527–530.

    MathSciNet  Google Scholar 

  63. Kovalevsky, J., 1968, “Review of Some Methods of Programming of Literal Developments in Celestial Mechanics”, Astron. J., 73, 203–209.

    Article  Google Scholar 

  64. Krylov, N. and Bogoliubov, N. N., 1947, “Introduction to Nonlinear Mechanics”, Ann. Math. Stud., 11, Princeton. Univ. Press, Princeton, New Jersey.

    Google Scholar 

  65. Kustaanheimo, P., 1964, “Spinor Regularization of Kepler Motion”, Ann. Univ. Turkuensis, A73, 3–7.

    MathSciNet  Google Scholar 

  66. Kyner, W. T., 1961, “invariant Manifolds”, Rend. Circ. Mat. Palermo, 10, 98–110.

    Article  MathSciNet  Google Scholar 

  67. Kyner, W. T., 1963, “A Mathematical Theory of the Orbits about an Oblate Planet”, Tech. Rep. to ONR, Det. Math., Univ. of Southern Calif., Los Angeles (51 pp.).

    Google Scholar 

  68. Kyner, W. T., 1968, “Rigorous and Formal Stability of Orbits about an Oblate Planet”, Mem. Amer. Math. Soc, 81, 1–27.

    MathSciNet  Google Scholar 

  69. Laricheva, V. V., 1966, “On Averaging a Certain Class of Systems of Nonlinear Differential Equations”, Diff. Equa., 2, No. 1, 169–173.

    Google Scholar 

  70. Lefschetz, S., 1959, “Differential Equations. Geometric Theory”, Interscience, New York.

    Google Scholar 

  71. Leimanis, E. and Minorsky, N., 1958, “Dynamics and Nonlinear Mechanics”, (Ch. I), J. Wiley, New York.

    MATH  Google Scholar 

  72. Levi-Civita, T., 1903, “Traiettorie Singolari ed Urti nel Problema Ristretto dei Tre Carpi”, Ann. Math., 9, 1–27.

    Google Scholar 

  73. Lindstedt, A., 1882, “Beitrag zur Integration der Differentialgleichungen der Störungtheorie”, Abh. K. Akad. Wiss. St. Petersburg, 31, No. 4.

    Google Scholar 

    Google Scholar 

  74. MacMillan, W. D., 1912, “A Method of Determining Solutions of a System of Analytic Functions in the Neighborhood of a Branch Point”, Math. Annalen, 72, 180.

    Article  MATH  Google Scholar 

  75. MacMillan, W. D., 1920, “Dynamics of Rigid Bodies”, Dover Publ., New York (pp. 403–413).

    Google Scholar 

  76. Mersman, W. A., 1971, “Explicit Recursive Algorithms for the Construction of Equivalent Canonical Transformations”, Cel. Mech., 3, 384–389.

    Article  MATH  MathSciNet  Google Scholar 

  77. Minorsky, N., 1962, “Nonlinear Oscillations”, van Nostrand, Princeton, New Jersey.

    MATH  Google Scholar 

  78. Moser, J., 1955, “Nonexistence of Integrals for Canonical Systems of Differential Equations”, Comm. Pure Appl. Math., 8, 409–436.

    Article  MATH  MathSciNet  Google Scholar 

  79. Moser, J., 1958, “New Aspects in the Theory of Stability of Hamiltonian Systems”, Comm. Pure Appl. Math., 2, 81–114.

    Article  Google Scholar 

  80. Moser, J., 1961, “A New Technique for the Construction of Solutions of Nonlinear Differential Equations”, Proc. Nat. Acad. Sci., 47, 1824–1831.

    Article  MATH  MathSciNet  Google Scholar 

  81. Moser, J., 1962, “On Invariant Curves of Area-Preserving Mappings of an Annulus”, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 1–20.

    Google Scholar 

  82. Moser, J., 1963, “Perturbation Theory for Almost Periodic Solutions for Undamped Nonlinear Differential Equations”, Int. Symp. on Nonlinear Diff. Eq. and Nonlinear Mech., Colorado Springs, 1961, Acad. Press, New York (pp. 71–79).

    Google Scholar 

  83. Moser, J., 1964, “Hamiltonian Systems”, Lecture Notes, New York Univ., New York.

    Google Scholar 

  84. Moser, J., 1966, “A Rapidly Convergent Iteration Method and Nonlinear Partial Differential Equations”, I, Ann. Scu. Norm. Sup. Pisa, 20, 265–315;

    Google Scholar 

  85. Moser, J., 1966, “A Rapidly Convergent Iteration Method and Nonlinear Partial Differential Equations”, II, Ann. Scu. Norm. Sup. Pisa, 20, 499–533.

    Google Scholar 

  86. Moser, J., 1966, “On the Theory of Quasi-Periodic Motions”, SIAM Rev., 8, 145–172.

    Article  MATH  MathSciNet  Google Scholar 

  87. Moser, J., 1967, “Convergent Series Expansions of Quasi-Periodic Motions”, Math. Ann., 169, 136–176.

    Article  MATH  MathSciNet  Google Scholar 

  88. Moser, J., 1970, “Regularization of Kepler’s Problem and the Averaging Method on a Manifold”, Comm. Pure Appl. Math., 23, 609–636.

    Article  MATH  MathSciNet  Google Scholar 

  89. Moulton, F. R., 1913, “Periodic Oscillating Satellites”, Math. Ann., 73, 441–479.

    Article  MathSciNet  Google Scholar 

  90. Moulton, F. R., 1920, “Periodic Orbits”, Carnegie Inst. Washington Publ., 161, Washington, D. C.

    Google Scholar 

  91. Musen, P., 1965, “On the high order effects in the Methods of Krylov-Bogoliubov and Poincaré”, J. Astronaut. Sci., 12, 129–134.

    Google Scholar 

  92. Nemitskii, V. V. and Stepanov, V. V., 1960, “Qualitative Theory of Differential Equations”, Princeton Univ. Press, Princeton, New Jersey.

    Google Scholar 

  93. Pliss, V. A., 1966, “On the Theory of Invariant Surfaces”, Diff. Eq., 2, 1139–1150.

    MathSciNet  Google Scholar 

  94. Pliss, V. A., 1966, “Nonlocal Problems of the Theory of Oscillations”, Acad. Press, New York.

    MATH  Google Scholar 

  95. Poincaré, H., 1893, “Les Méthodes Nouvelles de la Mécanique Céleste”, Vol. 2, Reprint by Dover Publ. New York (1957).

    MATH  Google Scholar 

  96. Poincaré, H., 1899, “Les Methodes Novelles de la Mécanique Céleste”, (Vol. 3), Gauthier-Viliars, Paris (Dover Publ. Reprint, New York).

    Google Scholar 

  97. Poincaré, H., 1909, “Léçons de Mécanique Céleste”, (Vol. 2), Gauthier-Villars, Paris.

    MATH  Google Scholar 

  98. Roels, J. and Louterman, G., 1970, “Normalization des Systèmes Linèaires Canoniques et Application au Problème Restreinte des Trois Corps”, Cel. Mech., 3, 129–140.

    Article  MATH  MathSciNet  Google Scholar 

  99. Sansone, G. and Conti, R., 1964, “Nonlinear Differential Equations”, Pergamon Press, New York.

    Google Scholar 

  100. Siegel, C. L., 1941, “On the Integrals of Canonical Systems”, Ann. Math. 42, 806–822.

    Article  Google Scholar 

  101. Siegel, C. L., 1954, “Über die Existenz einer Normal form analytischer Hamiitonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung” Math. Am., 128, 144–170.

    MATH  Google Scholar 

  102. Siegel, C. L., 1956, “Vorlesungen über Himmelsmechanik”, Springer-Verlag, Berlin.

    Google Scholar 

  103. Sternberg, S., 1969, “Celestial Mechanics”, (2. Vols.), W. A. Benjamin, New York.

    Google Scholar 

  104. Szebehely, V. et al., 1970, “Mean Motions and Characteristic Exponents at the Libration Points”, Astron. J., 75, 92–95.

    Article  Google Scholar 

  105. Volosov, V. M., 1962, “Averaging in Systems of Ordinary Differential Equations”, Russian Math. Surv., 17, 1–126.

    Article  MATH  MathSciNet  Google Scholar 

  106. Whittaker, E. T., 1961, “On the Adelphic Integrals of the Differential Equations of Dynamics”, Proc. Roy. Soc. Edinburg, 37, 95.

    Google Scholar 

  107. Whittaker, E. T., 1937, “A Treatise on the Analytical Dynamics of Particles and Rigid Bodies”, Cambridge, Univ, Press, London.

    MATH  Google Scholar 

  108. Wintner, A., 1947, “The Analytical Foundations of Celestial Mechanics”, Princeton Univ. Press, Princeton, New Jersey.

    MATH  Google Scholar 

  109. Zeipel, H. von, 1916–17, “Recherches sur le Mouvement des Petits Planets”, Arkiv. Astron. Mat. Phys., 11, 12, 13.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Sao Paulo, Brazil

    Georgio Eugenio Oscare Giacaglia

  2. University of Texas at Austin, USA

    Georgio Eugenio Oscare Giacaglia

Authors
  1. Georgio Eugenio Oscare Giacaglia
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1972 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Giacaglia, G.E.O. (1972). Perturbation Methods for Hamiltonian Systems. Generalizations. In: Perturbation Methods in Non-Linear Systems. Applied Mathematical Sciences, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6400-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-6400-2_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-90054-4

  • Online ISBN: 978-1-4612-6400-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation