Gyldén systems: Rotation of pericenters. (English) Zbl 0681.70015
Summary: A canonical transformation in phase space and a rescaling of time are proposed to reduce a Keplerian system with a time-dependent Gaussian parameter to a perturbed Keplerian system with a constant Gaussian parameter. When the time variation is slow, the perturbation through second order in the reduced problem is conservative, and, as a result, the orbital space of the averaged system in a sphere on which the phase flow causes a differential rotation representing a circulation of the line of apsides. The flow presents two isolated singularities corresponding to circular orbits travelled respectively in the direct and in the retrograde sense, and a degenerate manifold of fixed points corresponding to the collision orbits. Normalization beyond order two does not break the degeneracy. Adiabatic invariants, which are conservative functions, may be computed from the normalized Hamiltonian evaluated here to the fourth order. Nonetheless so high an approximation gives little information because the normalizing Lie transformation depends explicitly on the time through mixed secular-periodic terms. As an application, an estimate is offered for the apsidal rotation that a second order time derivative in the mass of the sun would induce on planetary orbits. This suggests an observational method for determining the latter parameter for the solar wind, but the predicted motions are too slow for the current level of observational precision.
MSC:
70F15 | Celestial mechanics |
70H15 | Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics |
70Sxx | Classical field theories |
70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
Keywords:
canonical transformation; phase space; rescaling of time; Keplerian system with a time-dependent Gaussian parameter; perturbed Keplerian system with a constant Gaussian parameter; Adiabatic invariants; normalized Hamiltonian; normalizing Lie transformation; mixed secular- periodic termsReferences:
[1] | Deprit, A. 1969,Celes. Mech. 1, 12-30. · Zbl 0172.26002 · doi:10.1007/BF01230629 |
[2] | Deprit, A. 1982,Celes. Mech. 26, 9-21. · Zbl 0512.70016 · doi:10.1007/BF01233178 |
[3] | Deprit, A. 1983,Celes. Mech. 31, 1-22. · Zbl 0545.70010 · doi:10.1007/BF01272557 |
[4] | Deprit, A. 1988,Celes. Mech., submitted. |
[5] | Feldman, W. C., Asbridge, J. R., Bame, S. J., and Gosling, J. T. 1977, inThe Solar Output and Its Variation, ed. O. R. White, (Boulder: Colorado Associated University Press), 351-382. |
[6] | Fock, V. 1959,The Theory of Space Time and Gravitation, (London: Pergamon Press). · Zbl 0085.42301 |
[7] | Gylden, H. 1884,Astron. Nachr. 109, No. 2593-94, col. 1-6. |
[8] | Hadjidemetriou, J. D. 1963,Icarus 2, 440-451. · doi:10.1016/0019-1035(63)90072-1 |
[9] | Hadjidemetriou, J. D. 1966,Icarus 5, 34-46. · doi:10.1016/0019-1035(66)90006-6 |
[10] | Lamport, L. 1986,LATEX. A Document Preparation System. (Reading, MA: Addison-Wesley Publishing Company). · Zbl 0852.68115 |
[11] | Macsyma Reference Manual 1986 (Cambridge, MA: Symbolics). |
[12] | Messchersky, J. 1902,Astron. Nachr. 159, No. 3807, col. 229-242. |
[13] | Miller, B. R. and Deprit, A. 1986,Proceedings of the Conference on Computers and Algebra, ed. J. Davenport, (New York: Springer-Verlag), in press. |
[14] | Misner C. W., Thorne K. S. and Wheeler, J. A. 1973,Gravitation, (San Francisco: Freeman and Company). |
[15] | Rom, A. 1970,Celes. Mech. 1, 301-319. · Zbl 0193.15203 · doi:10.1007/BF01231135 |
[16] | Saari, D. 1977,Celes. Mech. 16, 407-410. · Zbl 0381.70020 · doi:10.1007/BF01229284 |
[17] | Salmassi, M. 1985,Celes. Mech. 37, 359-370. · Zbl 0613.70003 · doi:10.1007/BF01261625 |
[18] | Sonett, C. 1974, inSolar Wind Three, ed. C. T. Russell, (Los Angeles: Institute of Geophysics and Planetary Physics, U.C.L.A.), 36. |
[19] | Van Flandern, T. C. 1976,Sci. Am. 234(2), 44-52. |
[20] | Vinti, J. P. 1974,Mon. Not. R. Astr. Soc. 169, 417-427. |
[21] | Vinti, J. P. 1977,Celest. Mech. 16, 391-406. · Zbl 0381.70019 · doi:10.1007/BF01229283 |
[22] | Wintner, A. 1947,The Analytical Foundations of Celestial Mechanics, (Princeton N. J.: Princeton University Press). · Zbl 0041.59006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.