Abstract
Mainfolds of orbits with stationary perigees are intrinsic features of the averaged main problem in aritifical satellite theory: they bifurcate off the manifold of circular orbits at the points where stability flips to instability and vice-versa.
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References
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First proposed by K.R. Meyer in Dynamical Systems, ed. M.M. Peixoto, Academic Press, New York, 259–272 (1973), and a year later by J. Marsden and A. Weinstein in Rep. Math. Phys. 5, 121–130(1974).The equivalent of Meyer’s reduction in quantum physics being the quantization procedure of Kirillov, Souriau, and Kostant, text-books in physics have taken to designate the classical construction as the KSK-reduction.
As expressed in A., Deprit, Celest. Mech., 26, 9–21 (1982), the concept of a Delaunay normalization got justified in intrinsic terms by R. Cushman in Differential Geometric Methods in Mathematical Physics, ed. S. Sternberg, D. Reidel Publishing Company, Dordrecht, 125–144(1984).
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These facts brought up first by H.G.L. Krause were later confirmed by R.E. Roberson, D. Brouwer, Astron. J. 63, 133–138(1958), and also, although somewhat inadvertently, by P. Herget and P. Musen, Astron.J. 63, 430–433 (1959).
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In regard to the origin and development of this LISP-based algebraic processor the reader is referred to a retrospective note by MACSYMA’s principal architect, J. Moses, in the Proceedings of the EUROSAM Conference ACM SIGSAM Bulletin, August 1974. A good introduction to the use of MACSYMA in nonlinear mechanics is to be found in R.H. Rand. Computer Algebra in Applied Mathematics. An Introduction to MACSYMA, Pitnam Advanced Publishing Program. Boston, MA (1984). As an illustration of its application to artificial satellite theory, see E. Zeis, A Computerized Algebraic Utility for the Construction of Non Singular Satellite Theory, M.S. Thesis Massachusetts Institute of Technology, Cambridge, MA (1978). Part of the thesis is published in E.Zeis, and P. Cefola., J.Guidance and Control 3, 48–54 (1978).
Our results complete the analysis begun by S. Aoki, Astron. J.67, 571–572 (1962), 68, 271–272, 355–365, 365–381 (1963),and continued by A.Jupp. Celest. Mech. 11, 361–378(1975) and 21, 361–393(1980).
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Coffey, S.L., Deprit, A., Miller, B.R. (1986). The Nature of the Critical Inclinations in Artificial Satellite Theory. In: Bhatnagar, K.B. (eds) Space Dynamics and Celestial Mechanics. Astrophysics and Space Science Library, vol 127. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4732-0_4
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DOI: https://doi.org/10.1007/978-94-009-4732-0_4
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