Reduction of polynomial Hamiltonians by the construction of formal integrals

Nonlinearity

Volumn 13, Issue 4, 2000, Pages 1021-1054

  Palacián, Jesús ^a   Yanguas, Patricia ^a  

^a PUBLIC UNIVERSITY OF NAVARRE   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034216023     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/13/4/303     Document Type: Article

References (52)

1. Arms J M, Cushman R H and Gotay M J 1991 A universal reduction procedure for Hamiltonian group actions The Geometry of Hamiltonian Systems (Berkeley, 1989) ed T Ratiu (Berlin: Springer) pp 33-51
2. Arnold V I 1978 Ordinary Differential Equations (Cambridge, MA: MIT Press)
3. Arnold V I, Kozlov V V and Neishtadt A I 1988 Mathematical aspects of classical and celestial mechanics Dynamical Systems vol 3 (Encyclopaedia of Mathematical Sciences vol 3) ed V I Arnold (Berlin: Springer)
4. Birkhoff G D 1927 Dynamical Systems (AMS Coll. Publ. vol 9) (Providence, RI: American Mathematical Society)
5. Broer H W, Chow S-N, Kim Y-I and Vegter G 1993 A normally elliptic Hamiltonian bifurcation Z. Angew. Math. Phys. 44 389-432
6. Broer H W, Chow S-N, Kim Y-I and Vegter G 1995 The Hamiltonian double-zero eigenvalue Normal Forms and Homoclinic Chaos (Fields Institute Communications vol 4) ed W F Langford and W Nagata pp 1-19
7. Celletti A and Giorgilli A 1991 On the stability of the Lagrangian points in the spatial restricted problem of three bodies Celest. Mech. Dyn. Astron. 50 31-58
8. Churchill R C, Kummer M and Rod D L 1983 On averaging, reduction and symmetry in Hamiltonian systems J. Differ. Equ. 49 359-414
9. Cicogna G and Gaeta G 1994 Normal forms and nonlinear symmetries J. Phys. A: Math. Gen. 27 7115-24
10. Coddington E A and Levinson N 1955 Theory of Ordinary Differential Equations (New York: McGraw-Hill)
11. Cushman R, Deprit A and Mosak R 1983 Normal form and representation theory J. Math. Phys. 24 2102-17
12. de Zeeuw T 1985 Motion in the core of a triaxial potential Mon. Not. R. Astron. Soc. 215 731-60
13. Deprit A 1969 Canonical transformations depending on a small parameter Celest. Mech. 1 12-30
14. Deprit A 1982 Delaunay normalisations Celest. Mech. 26 9-21
15. Deprit A, Palacián J, Deprit E and San Juan J F 1999 The relegation algorithm Celest. Mech. Dyn. Astron. submitted
16. Duistermaat J J 1984 Non-integrability of the 1:1:2-resonance Ergod. Theor. Dyn. Syst. 4 553-68
17. Elphick C, Tirapegui E, Brachet M E, Coullet P and Iooss G 1987 A simple global characterization for normal forms of singular vector fields Physica D 29 95-127
18. Giorgilli A 1979 A computer program for integrals of motion Comput. Phys. Commun. 16 331-43
19. Giorgilli A and Galgani L 1978 Formal integrals for an autonomous Hamiltonian system near an equilibrium point Celest. Mech. 17 267-80
20. Gustavson F G 1966 On constructing formal integrals of a Hamiltonian system near an equilibrium point Astron. J. 71 570-86
21. Haller G and Wiggins S 1996 Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems Physica D 90 319-65
22. Hénon M and Heiles C 1964 The applicability of the third integral of motion: some numerical experiments Astron. J. 69 73-9
23. Henrard J 1970 On a perturbation theory using Lie transforms Celest. Mech. 3 107-20
24. Hoveijn I 1993 Aspects of resonance in dynamical systems PhD Thesis Universiteit Utrecht
25. Hoveijn I and Verhulst F 1990 Chaos in the 1:2:3 Hamiltonian normal form Physica D 44 397-406
26. Jorba A 1999 A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems Exp. Math. 8 155-95
27. Jorba A and Villanueva J 1997 On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems Nonlinearity 10 783-822
28. Kamel A A 1969 Expansion formulae in canonical transformations depending on a small parameter Celest. Mech. 1 190-9
29. Kummer M 1976 On resonant non linearly coupled oscillators with two equal frequencies Commun. Math. Phys. 48 53-79
30. Kummer M 1978 On resonant classical Hamiltonians with two equal frequencies Commun. Math. Phys. 58 85-112
31. Laub A J and Meyer K 1974 Canonical forms for symplectic and Hamiltonian matrices Celest. Mech. 9 213-38
32. Lichtenberg A J and Lieberman M A 1992 Regular and Chaotic Dynamics 2nd edn (Applied Mathematical Sciences vol 38) (New York: Springer)
33. Meyer K R 1974 Normal forms for Hamiltonian systems Celest. Mech. 9 517-22
34. Meyer K R 1984 Normal forms for the general equilibrium Funkcial. Ekva. 27 261-71
35. Meyer K R 1991 A Lie transform tutorial II Computer Aided Proofs in Analysis vol 28 (The IMA Volumes in Mathematics and its Applications) ed K R Meyer and D S Schmidt (New York: Springer) pp 190-210
36. Meyer K R and Hall G R 1992 Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (Applied Mathematical Sciences vol 90) (New York: Springer)
37. Meyer K R and Schmidt D S 1971 Periodic orbits near ℒ4 for mass ratios near the critical mass ratio of Routh Celest. Mech. 4 99-109
38. Mikram J and Zinoun F 1999 Computation of normal forms of Hamiltonian systems in the presence of Poisson commuting integrals Numer. Algorithms 21 287-310
39. Moser J 1958 New aspects in the theory of stability of Hamiltonian systems Commun. Pure Appl. Math. 11 81-114
40. Moser J 1970 Regularization of Kepler's problem and the averaging method on a manifold Commun. Pure Appl. Math. 23 609-36
41. Palacián J 1993 Teoría del satélite artificial: armónicos teserales y su relegación mediante simplificaciones algebraicas PhD Thesis Publicaciones del Seminario Matemático García de Galdeano, Serie II Sección 2 vol 45 Universidad de Zaragoza, Zaragoza
42. Palacián J and Yanguas P 1999 Simplification of perturbed Hamiltonians through Lie transformations Proc. 3rd Int. Symp. HAMSYS'98 on Geometrical Methods in Hamiltonian Systems and Celestial Mechanics (Pátzcuaro) accepted
43. Palacián J and Yanguas P 2000 Generalization of normal forms for polynomial vector fields J. Math. Pure Appl. submitted
44. Poincaré M H 1885 Sur les courbes définies par les équations différentielles J. Math. Pure Appl. IV 1 167-244
45. Sanders J A and Verhulst F 1985 Averaging Methods in Nonlinear Dynamical Systems (Applied Mathematical Sciences vol 59) (New York: Springer)
46. Sokol'skii A G 1974-5 On the stability of an autonomous Hamiltonian system with two degrees of freedom in the case of equal frequencies J. Appl. Math. Mech. 38 741-9
47. van der Meer J-C 1982 Nonsemisimple 1:1 resonance at an equilibrium Celest. Mech. 27 131-49
48. van der Meer J-C 1985 The Hamiltonian Hopf Bifurcation (Lecture Notes in Mathematics vol 1160) (New York: Springer)
49. Whittaker E T 1927 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge: Cambridge University Press)
50. Wiggins S 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos (Texts in Applied Mathematics vol 2) (Berlin: Springer)
51. Wolfram S 1996 The Mathematica Book 3rd edn (Cambridge: Wolfram Media/Cambridge University Press)
52. Yanguas P 1998 Integrability, normalization and symmetries of Hamiltonian systems in 1-1-1 resonance PhD Thesis Publicaciones del Departamento de Matemática e Informática, Sección 2 vol 10 Universidad Pública de Navarra, Pamplona