Abstract
In this paper, the relationship between the Dragt-Finn transform and the classical Lie transform introduced by Deprit is discussed. The relative performance of the algorithms used for the computations of the transformed functions is compared, and the relation between their generators is given. These generators produce the same transform which insures the construction of the same invariants.
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References
G.E. Andrews,The Theory of Partitions, Encyclopedia of Mathematics, Number Theory, Vol. 2, Addison-Wesley, 1976.
V.I. Arnold,Mathematical Methods for Classical Mechanics, Graduate texts in Mathematics, Vol. 60, Springer-Verlag, 1978.
J.R. Cary,Physics Reports, North-Holland Publishing Company.
A. Deprit, ‘Canonical Transformations Depending on a Small Parameter’,Celestial Mechanics 1 (1969), 12–30.
J. Dragt, J. M. Finn, ‘Lie Series and Invariant Functions for Analytic Symplectic Maps’,Journal of Mathematical Physic 17 (1976), 2215–2227.
F. Fassò, ‘On a Relation among Lie Series’,Celestial Mechanics and Dynamical Astronomy 46 (1989), 113–118.
J.M. Finn, ‘Lie Series: a Perspective, Local and Global Methods of nonlinear Dynamics’,Lecture Notes in Physics 252 (1984), 63–86.
E. Fried, G. Ezra, ‘PERTURB: A Special-Purpose Algebraic Manipulation Program for Classical Perturbation Theory’,Journal of Computational Chemistry (1987), 397–411.
A. Giorgilli, ‘Rigourous Results on the Power Expansions for the Integrals of a Hamiltonian System near a Elliptic Equilibrium Point’,Annales de l'Institut Henri Poincaré 48 (1988), 423–439.
A. Giorgilli, L. Galgani, ‘Formal Integrals for an Autonomous Hamiltonian System Near an Equilibrium Point’,Celestial Mechanics 17 (1977), 267–280.
Giorgilli, Delshams, Fontich, Galgani, Simò, ‘Effective Stability for a Hamiltonian System near an Elliptic Equilibrium, with an Application to the restricted Three Body Problem’,Journal of Diff. Equations (1988).
J. Henrard, ‘The Algorithm of the Inverse for Lie Transform’,Recent Advances in Dynamical Astronomy (1973), 250–259.
P.-V. Koseleff, ‘Relations among Formal Lie Series and Construction of Symplectic Integrators’,AAECC' 10 proceedings, in print (1993).
M.A. Lichtenberg, A.J. Lieberman, Applied Mathematical Sciences 92, Springer-Verlag, 1988.
P. Lochak, C. Meunier,Multiphase Averaging for Classical Systems, Applied Mathematical Sciences 78, Springer-Verlag, New York, 1988.
K. R. Meyer, G. R. Hall,Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences 92, Springer Verlag New-York, 1992.
J. Michel, ‘Bases des Algèbres de Lie et Série de Hausdorff’,Séminaire Dubreil, Paris 27, 6 (1974).
C. Simò, ‘Estabilitat de Sistemes Hamiltonians’,Memorias de la Real Academia de Ciencias y Artes de Barcelona 48 (1989), 303–348.
S. Steinberg, ‘Lie Series, Lie Transformations, and their Applications’, inLie Methods in Optics, Lecture Notes in Physics 250 (1985).
G. Viennot,Algèbres de Lie libres et Monoïdes Libres, Lecture Notes in Mathematics, vol 691, Springer Verlag, 1978.
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Koseleff, P.V. Comparison between Deprit and Dragt-Finn perturbation methods. Celestial Mech Dyn Astr 58, 17–36 (1994). https://doi.org/10.1007/BF00692115
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DOI: https://doi.org/10.1007/BF00692115