Theoretical Chemistry Accounts

, 133:1571 | Cite as

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

  • Hiroshi Teramoto
  • Mikito Toda
  • Tamiki Komatsuzaki
Regular Article
Part of the following topical collections:
  1. Ezra Festschrift Collection


Validity ranges of Lie canonical perturbation theory (LCPT) are investigated in terms of non-blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non-blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non-blow-up regions enclose the region inside the separatrix of the Hamiltonian, but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non-blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix, whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non-blow-up regions do not necessarily cover the whole regions inside the HCN well. We propose a new perturbation method to improve non-blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in Teramoto and Komatsuzaki (J Chem Phys 129:094302, 2008; Phys Rev E 78:017202, 2008).


Lie canonical perturbation theory Non-blow-up region  HCN isomerization 



We would like to dedicate this article to continuous stimulating and pioneering works by Professor Greg. Ezra in the research field of classical and semiclassical chemical dynamics. HT would like thank Professor Kazuyuki Yagasaki and Zin Arai for their valuable comments on the definition of the action in the Hamiltonian with one degree of freedom and Professor Turgay Uzer for his comment on the Padé approximation as an alternative method for improving the validity range. This work has been supported by JSPS, the Cooperative Research Program of “Network Joint Research Center for Materials and Devices”, Research Center for Computational Science, Okazaki, Japan, and Grant-in-Aid for challenging Exploratory Research (to TK), and Grant-in-Aid for Scientific Research (B) (to TK) from the Ministry of Education, Culture, Sports, Science and Technology, and Nara Women’s University Intramural Grant for Project Research (to MT), Grant-in-Aid for challenging Exploratory Research (to MT), and Grant-in-Aid for Scientific Research (C) (to MT) from the Ministry of Education, Culture, Sports, Science and Technology.


  1. 1.
    Teramoto H, Komatsuzaki T (2008) Exploring remnant of invariants buried in a deep potential well in chemical reactions. J Chem Phys 129:094302CrossRefGoogle Scholar
  2. 2.
    Teramoto H, Komatsuzaki T (2008) Probing remnants of invariant s to mediate energy exchange in highly-chaotic many-dimensional systems. Phys Rev E 78:017202CrossRefGoogle Scholar
  3. 3.
    Lichtenberg AJ, Lieberman MA (1991) Regular and chaotic dynamics, 2nd edn. Springer, New YorkGoogle Scholar
  4. 4.
    Cary JR (1981) Lie transform perturbation theory for Hamiltonian systems. Phys Rev 79:129Google Scholar
  5. 5.
    Hori G (1966) Theory of general perturbations with unspecified canonical variables. Publ Astron Soc Jpn 18:287Google Scholar
  6. 6.
    Hori G (1967) Non-linear coupling of two harmonic oscillations. Publ Astron Soc Jpn 19:229Google Scholar
  7. 7.
    Deprit A (1969) Canonical transformations depending on a small parameter. Celest Mech 1:12CrossRefGoogle Scholar
  8. 8.
    Dragt AJ, Finn JM (1976) Lie series and invariant functions for analytic symplectic maps. J Math Phys 17:2215CrossRefGoogle Scholar
  9. 9.
    Campbell JA, Jefferys WH (1970) Equivalence of the perturbation theories of Hori and Deprit. Celest Mech 2:467CrossRefGoogle Scholar
  10. 10.
    Marsman WA (1970) A new algorithm for the Lie transformation. Celest Mech 3:81CrossRefGoogle Scholar
  11. 11.
    Koseleff PV (1994) Comparison between Deprit and Dragt-Finn perturbation methods. Celest Mech Dyn Astron 58:17CrossRefGoogle Scholar
  12. 12.
    Murdock J (2003) Normal forms and unfoldings for local dynamical systems. Springer monographs in mathematics, 1st edn. Springer, New YorkCrossRefGoogle Scholar
  13. 13.
    Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems. Applied mathematical sciences, 2nd edn. Springer, New YorkGoogle Scholar
  14. 14.
    Broer H, Hoveijn I, Lunter G, Vegter G (2003) Bifurcations in Hamiltonian Systems. Lecture notes in mathematics, vol 1806. Springer, BerlinGoogle Scholar
  15. 15.
    Siegel CL (1941) On the integrals of canonical systems. Ann Math 42:806CrossRefGoogle Scholar
  16. 16.
    Bryuno AD (1975) Normal form of real differential equations. Math Notes 18:722CrossRefGoogle Scholar
  17. 17.
    Bryuno AD (1982) Divergence of a real normalizing transformation. Math Notes 31:207CrossRefGoogle Scholar
  18. 18.
    Ito H (1989) Convergence of Birkhoff normal forms for integrable systems. Comment Math Helv 64:412CrossRefGoogle Scholar
  19. 19.
    Ito H (1992) Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Math Ann 292:411CrossRefGoogle Scholar
  20. 20.
    Bruno AD, Walcher S (1994) Symmetries and convergence of normalizing transformations. J Math Anal Appl 183:571CrossRefGoogle Scholar
  21. 21.
    Cicogna G (1996) On the convergence of normalizing transformations in the presence of symmetries. J Math Anal Appl 199:243CrossRefGoogle Scholar
  22. 22.
    Kappeler T, Kodama Y, Némethi A (1998) On the Birkhoff normal form of a completely integrable Hamiltonian system near a fixed point with resonance. Ann Scuola Norm Sup Pisa Cl Sci XXVI:623Google Scholar
  23. 23.
    Walcher S (2000) On convergent normal form transformations in presence of symmetries. J Math Anal Appl 244:17CrossRefGoogle Scholar
  24. 24.
    Pérez-Marco P (2001) Total convergence or general divergence in small divisors. Commun Math Phys 223:451CrossRefGoogle Scholar
  25. 25.
    Cicogna G, Walcher S (2002) Convergence of normal form transformations: the role of symmetries. Acta Appl Math 70:95CrossRefGoogle Scholar
  26. 26.
    Zung NT (2005) Convergence versus integrability in Birkhoff normal form. Ann Math 161:141CrossRefGoogle Scholar
  27. 27.
    Chiba H (2009) Extension and unification of singular perturbation methods for ODEs based on the renormalization group method. SIAM J Appl Dyn Syst 8:1066CrossRefGoogle Scholar
  28. 28.
    Markus L, Meyer KR (1974) Generic hamiltonian dynamical systems are neither integrable nor ergodic. Mem Am Math Soc 144Google Scholar
  29. 29.
    Koon WS, Lo MW, Marsden JE, Ross SD (2000) Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10:427CrossRefGoogle Scholar
  30. 30.
    Jaffe C, Ross SD, Lo MW, Marsden J, Farrelly D, Uzer T (2002) Statistical theory of asteroid escape rates. Phys Rev Lett 89:011101CrossRefGoogle Scholar
  31. 31.
    von Milczewski J, Diercksen GHF, Uzer T (1996) Computation of the Arnol’d Web for the hydrogen atom in crossed electric and magnetic fields. Phys Rev Lett 76:2890CrossRefGoogle Scholar
  32. 32.
    Uzer T, Jaffé C, Palacián J, Yanguas P, Wiggins S (2002) The geometry of reaction dynamics. Nonlinearity 15:957CrossRefGoogle Scholar
  33. 33.
    Komatsuzaki T, Berry RS (1999) Regularity in chaotic reaction paths. \(\text{ I }.\, \text{ Ar }_6\). J Chem Phys 110:9160–9173CrossRefGoogle Scholar
  34. 34.
    Komatsuzaki T, Berry RS (1999) Regularity in chaotic reaction path \(\text{ II }:\, \text{ Ar }_6\)—energy dependence and visualization of the reaction bottleneck. Phys Chem Chem Phy. 1:1387CrossRefGoogle Scholar
  35. 35.
    Komatsuzaki T, Berry RS (2000) Local regularity and non-recrossing path in transition states—a new strategy in chemical reaction theories. J Mol Struct (Theochem) 506:55CrossRefGoogle Scholar
  36. 36.
    Komatsuzaki T, Berry RS (2001) Regularity in chaotic reaction paths. III: local invariances at the reaction bottleneck. J Chem Phys 115:4105CrossRefGoogle Scholar
  37. 37.
    Komatsuzaki T, Berry RS (2001) Dynamical hierarchy in transition states: why and how does a system climb over the mountain? Proc Natl Acad Sci USA 98:7666CrossRefGoogle Scholar
  38. 38.
    Komatsuzaki T, Berry RS (2002) A dynamical propensity rule of transitions in chemical reactions. J Phys Chem A 106:10945CrossRefGoogle Scholar
  39. 39.
    Komatsuzaki T, Berry RS (2002) Chemical reaction dynamics: many-body chaos and regularity. Adv Chem Phys 123:79Google Scholar
  40. 40.
    Komatsuzaki T, Nagaoka M (1996) Study on “regularity” of the barrier recrossing motion. J Chem Phys 105:10838CrossRefGoogle Scholar
  41. 41.
    Komatsuzaki T, Nagaoka M (1997) A dividing surface free from a barrier recrossing motion in many-body systems. Chem Phys Lett 265:91CrossRefGoogle Scholar
  42. 42.
    Kawai S, Fujimura Y, Kajimoto O, Yamashita T, Li C-B, Komatsuzaki T, Toda M (2007) Dimension reduction for extracting geometrical structure of multidimensional phase space: application to fast energy exchange in the reaction \(\text{ O }(^1{{\rm D}})+\text{ N }_2{{\rm O}}\rightarrow \text{ NO }+\text{ NO }\). Phys Rev A 75:022714CrossRefGoogle Scholar
  43. 43.
    Kawai S, Komatsuzaki T (2010) Robust existence of a reaction boundary to separate the fate of a chemical reaction. Phys Rev Lett 105:048304CrossRefGoogle Scholar
  44. 44.
    Jaffé C, Kawai S, Palacián J, Yanguas P, Uzer T (2005) A new look at the transition state: Wigner’s dynamical perspective revisited. Adv Chem Phys 130:171Google Scholar
  45. 45.
    Li C-B, Matsunaga Y, Toda M, Komatsuzaki T (2005) Phase space reaction network on a multisaddle energy landscape: Hcn isomerization. J Chem Phys 123:184301CrossRefGoogle Scholar
  46. 46.
    Waalkens H, Burbanks A, Wiggins S (2004) Phase space conduits for reaction in multidimensional systems, HCN isomerization in three dimensions. J Chem Phys 121:6207CrossRefGoogle Scholar
  47. 47.
    Bartsch T, Hernandez R, Uzer T (2005) Transition state in a noisy environment. Phys Rev Lett 95:058301CrossRefGoogle Scholar
  48. 48.
    Bartsch T, Uzer T, Hernandez R (2005) Stochastic transition states: reaction geometry amidst noise. J Chem Phys 123:204102CrossRefGoogle Scholar
  49. 49.
    Bartsch T, Uzer T, Moix JM, Hernandez R (2006) Identifying reactive trajectories using a moving transition state. J Chem Phys 124:244310CrossRefGoogle Scholar
  50. 50.
    Kawai S, Komatsuzaki T (2009) Dynamical reaction coordinate buried in thermal fluctuation i: time-dependent normal form theory for multidimensional underdamped langevin equation. J Chem Phys 131:224505CrossRefGoogle Scholar
  51. 51.
    Kawai S, Komatsuzaki T (2009) Dynamical reaction coordinate buried in thermal fluctuation ii: numerical examples. J Chem Phys 131:224506CrossRefGoogle Scholar
  52. 52.
    Kawai S, Komatsuzaki T (2010) Hierarchy of reaction dynamics in a thermally fluctuating environment. Phys Chem Chem Phys 12:7626–7635CrossRefGoogle Scholar
  53. 53.
    Kawai S, Komatsuzaki T (2010) Nonlinear dynamical effects on reaction rate constants in thermally fluctuating environments. Phys Chem Chem Phys 12:7636–7647CrossRefGoogle Scholar
  54. 54.
    Kawai S, Komatsuzaki T (2010) Dynamical reaction coordinate in thermally fluctuating environment in the framework of multidimensional generalized langevin equations. Phys Chem Chem Phys 12:15382–15391CrossRefGoogle Scholar
  55. 55.
    Fried LE, Ezra GS (1987) Semiclassical quantization using perturbation theory: algebraic quantization of multidimensional systems. J Chem Phys 86:6270CrossRefGoogle Scholar
  56. 56.
    Fried LE, Ezra GS (1988) Perturb: a special-purpose algebraic manipulation program for classical perturbation theory. Comput Phys Commun 51:103CrossRefGoogle Scholar
  57. 57.
    Fried LE, Ezra GS (1988) Semiclassical quantization of polyatomic molecules: some recent developments. J Phys Chem 92:3144CrossRefGoogle Scholar
  58. 58.
    Kawai S, Komatsuzaki T (2011) Quantum reaction boundary to mediate reactions in laser fields. J Chem Phys 134:024317CrossRefGoogle Scholar
  59. 59.
    Kawai S, Komatsuzaki T (2012) Laser control of chemical reactions by phase space structures. Bull Chem Soc Jpn 85:854–861CrossRefGoogle Scholar
  60. 60.
    Giorgilli A, Galgani L (1985) Rigorous estimates for the series expansions of hamiltonian perturbation theory. Celest Mech 37:95CrossRefGoogle Scholar
  61. 61.
    Arnold V (1964) Instabilities in dynamical systems with several degrees of freedom. Sov Math Dokl 5:581Google Scholar
  62. 62.
    Chirikov BV (1979) A universal instability of many-dimensional oscillator systems. Phys Rep 52:263CrossRefGoogle Scholar
  63. 63.
    Guzzo M, Lega E, Froeschlé C (2009) A numerical study of the topology of normally hyperbolic invariant manifolds supporting arnold diffusion in quasi-integrable systems. Phys D 238:1797CrossRefGoogle Scholar
  64. 64.
    Cincottaa PM, Efthymiopoulosb C, Giordanoa CM, Mestrea MF (2014) Chirikov and nekhoroshev diffusion estimates: bridging the two sides of the river. Phys D 266:49CrossRefGoogle Scholar
  65. 65.
    Martens CC, Davis MJ, Ezra GS (1987) Local frequency analysis of chaotic motion in multidimensional systems: energy transport and bottlenecks in planar OCS. Chem Phys Lett 142:519CrossRefGoogle Scholar
  66. 66.
    Atkins KM, Logan DE (1992) Intersecting resonances as a route to chaos: classical and quantum studies of a three-oscillator model. Phys Lett A 162:255CrossRefGoogle Scholar
  67. 67.
    Froeschlé C, Guzzo M, Lega E (2000) Graphical evolution of the arnold web: from order to chaos. Science 289:2108CrossRefGoogle Scholar
  68. 68.
    Chandre C, Wiggins S, Uzer T (2003) Time-frequency analysis of chaotic systems. Phys D 181:171CrossRefGoogle Scholar
  69. 69.
    Shojiguchi A, Li C-B, Komastuzaki T, Toda M (2006) Wavelet analysis and Arnold web picture for detecting energy transfer in a Hamiltonian dynamical system. Laser Phys 17:1097CrossRefGoogle Scholar
  70. 70.
    Arnold VI, Kozlov VV, Neishtadt AI (2006) Mathematical aspects of classical and celestial mechanics. Encyclopedia of mathematical sciences, 3rd edn. Springer, BerlinGoogle Scholar
  71. 71.
    Laskar J (1993) Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Phys D 67:257CrossRefGoogle Scholar
  72. 72.
    Honjo S, Kaneko K (2003) Structure of resonances and transport in multidimensional Hamiltonian dynamical systems. Adv Chem Phys 130B:437Google Scholar
  73. 73.
    Semparithi A, Keshavamurthy S (2006) Intramolecular vibrational energy redistribution as state space diffusion: classical-quantum correspondence. J Chem Phys 125:141101CrossRefGoogle Scholar
  74. 74.
    Shojiguchi A, Li C-B, Komastuzaki T, Toda M (2007) Fractional behavior in nonergodic reaction processes of isomerization. Phys Rev E75:035204(R)Google Scholar
  75. 75.
    Shojiguchi A, Li C-B, Komastuzaki T, Toda M (2007) Fractional behavior in multidimensional Hamiltonian systems describing reactions. Phys Rev E76:056205Google Scholar
  76. 76.
    Wiggins S (1990) On the geometry of transport in phase space I. Transport in k-degree-of-freedom Hamiltonian systems, \(2 \le k \le \infty\). Phys D 44:471CrossRefGoogle Scholar
  77. 77.
    Gillilan RE, Ezra GS (1991) Transport and turnstiles in multidimensional hamiltonian mappings for unimolecular fragmentation: application to van der Waals predissociation. J Chem Phys 94:2648CrossRefGoogle Scholar
  78. 78.
    Toda M (1995) Crisis in chaotic scattering of a highly excited van der waals complex. Phys Rev Lett 74:2670CrossRefGoogle Scholar
  79. 79.
    Shojiguchi A, Li C-B, Komastuzaki T, Toda M (2008) Dynamical foundation and limitation of statistical reaction theory. Commun Nonlinear Sci Numer Simul 13:857CrossRefGoogle Scholar
  80. 80.
    Goldstein H, Poole CP Jr, Safko JL (2001) Classical mechanics, 3rd edn. Addison-Wesley, BostonGoogle Scholar
  81. 81.
    Coddington EE (1984) Theory of ordinary differential equations. Krieger Pub Co, HuntingtonGoogle Scholar
  82. 82.
    Dragt AJ, Finn JM (1979) Normal form mirror machine hamiltonians. J Math Phys 20:2649CrossRefGoogle Scholar
  83. 83.
    Coleman CS (1984) Boundedness and unboundedness in polynomial differential systems. Nonlinear Anal Theory Methods Appl 8:1287CrossRefGoogle Scholar
  84. 84.
    Murrell JN, Carter S, Halonen LO (1982) Frequency optimized potential energy functions for the ground-state surfaces of hcn and hcp. J Mol Spectrosc 93:307CrossRefGoogle Scholar
  85. 85.
    Ahrens J, Geveci B, Law C (2005) ParaView: an end-user tool for large data visualization. In: Hansen C, Johnson C (eds) The visualization handbook. Academic Press, London, p 717CrossRefGoogle Scholar
  86. 86.
    Shirts RB, Reinhardt WP (1982) Approximate constants of motion for classically chaotic vibrational dynamics: vague tori, semiclassical quantization, and classical intramolecular energy flow. J Chem Phys 77:5204CrossRefGoogle Scholar
  87. 87.
    Ali MK, Wood WR, Devitt JS (1986) On the summation of the Birkhoff–Gustavson normal form of an anharmonic oscillator. J Math Phys 27:1806CrossRefGoogle Scholar
  88. 88.
    Ali MK, Wood WR (1987) The Birkhoff–Gustavson normal form of Double-Well anharmonic oscillators. Prog Theor Phys 78:766CrossRefGoogle Scholar
  89. 89.
    Robnik M (1993) On the Padé approximations to the Birkhoff–Gustavson normal form. J Phys A Math Gen 26:7427CrossRefGoogle Scholar
  90. 90.
    Li CB, Shojiguchi A, Toda M, Komatsuzaki T (2006) Definability of no-return transition states in high energy regime above threshold. Phys Rev Lett 97:028302CrossRefGoogle Scholar
  91. 91.
    Teramoto H, Takatsuka K (2007) Local integrals and their globally connected invariant structure in phase space giving rise to a promoting mode of chemical reaction. J Chem Phys 126:124110CrossRefGoogle Scholar
  92. 92.
    Baker GA Jr, Graves-Morris P (1996) Padé approximants, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  93. 93.
    Kaluža M, Robnik M (1992) Improved accuracy of the Birkhoff–Gustavson normal form and its convergence properties. J Phys A Math Gen 25:5311CrossRefGoogle Scholar
  94. 94.
    Arnold VI (1963) Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ Math Surv 18:9CrossRefGoogle Scholar
  95. 95.
    Howland RA (1977) An accelerated eliminations technique for the solution of perturbed Hamiltonian systems. Celest Mech 15:327CrossRefGoogle Scholar
  96. 96.
    Howland RA, Richardson DL (1984) The Hamiltonian transformation in quadratic Lie transforms. Celest Mech 32:99CrossRefGoogle Scholar
  97. 97.
    Gabern F, Jorba À, Locatelli U (2005) On the construction of the Kolmogorov normal form for the Trojan asteroids. Nonlinearity 18:1705CrossRefGoogle Scholar
  98. 98.
    Uzer T (1991) Theories of intramolecular vibrational energy transfer. Phys Rep 199:73CrossRefGoogle Scholar
  99. 99.
    Press WH, Teukolosky SA, Vetterling WT, Flannery BP (2007) Numerical recipes, the art of scientific computing. International series of monographs on chemistry, 3rd edn. Cambridge University Press, CambridgeGoogle Scholar
  100. 100.
    Strpistrup B (2008) Programming: principles and practice using C++, 3rd edn. Addison-Wesley Professional, BostonGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hiroshi Teramoto
    • 1
  • Mikito Toda
    • 2
  • Tamiki Komatsuzaki
    • 1
  1. 1.Molecule and Life Nonlinear Sciences Laboratory, Research Institute for Electronic ScienceHokkaido UniversitySapporoJapan
  2. 2.Nonequilibrium Dynamics Laboratory, Research group of Physics, Division of Natural ScienceNara Women’s UniversityNaraJapan

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