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A new algorithm for the Lie transformation

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Abstract

Following Hori, the Lie transformation is presented in a form that is independent of any extraneous parameters. The transformation is canonical, and its inverse is obtained by changing the sign of the generating function. The introduction of a small parameter into the generating function and the Hamiltonian then yields a recursive, triangular algorithm. The case of a Hamiltonian containing the time explicitly is included by adjoining an additional pair of conjugate variables. The necessary and sufficient condition that this transformation be identical to Deprit's transformation is given as a recursive relation between successive terms in the generating functions. Explicit formulas are obtained through the sixth order.

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References

  • Deprit, A.: 1969, ‘Canonical Transformations Depending on a Small Parameter’,Celestial Mech. 1, 12–30.

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  • Hori, G.-J.: 1966, ‘Theory of General Perturbations with Unspecified Canonical Variables’,Publ. Astron. Soc. Japan 18, 287–296.

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  • Kamel, A. A.: 1969, ‘Expansion Formulae in Canonical Transformations Depending on a Small Parameter’,Celestial Mech. 1, 190–199.

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  • Mersman, W. A.: 1969, ‘A Unified Treatment of Lunar Theory and Artificial Satellite Theory’, NASA Technical Note D-5459.

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Authors and Affiliations

  1. NASA Ames Research Center, Moffett Field, Calif., U.S.A.

    William A. Mersman

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  1. William A. Mersman
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Additional information

After submitting the present paper the author learned of similar and independent work by Campbell and Jefferys and by Kamel (Ph.D. thesis).

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Mersman, W.A. A new algorithm for the Lie transformation. Celestial Mechanics 3, 81–89 (1970). https://doi.org/10.1007/BF01230434

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  • DOI: https://doi.org/10.1007/BF01230434

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