Celestial Mechanics and Dynamical Astronomy

, Volume 98, Issue 4, pp 251–283 | Cite as

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

  • Michael Efroimsky
  • Alberto Escapa
Original Article


In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).


Earth rotation Attitude mechanics Attitude dynamics Kinoshita theory of the Earth rotation Kinoshita–Souchay theory of the Earth rotation Hamiltonian theory of the Earth rotation Andoyer variables Andoyer elemets Osculation Nonosculation Canonical perturbation theory Rigid-body rotation Right-body dynamics Rigid-body mechanics Poinsot problem Euler–Poinsot problem 


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  1. Abdullah K. and Albouy A. (2001). On a strange resonance noticed by M. Herman. Regular and Chaotic Dynamics 6: 421–432 MATHCrossRefGoogle Scholar
  2. Andoyer H. (1923). Cours de Mécanique Céleste. Gauthier-Villars, Paris MATHGoogle Scholar
  3. Boccaletti D. and Pucacco G. (2002). Theory of Orbits. Volume 2: Perturbative and Geometrical Methods, Chapter 8. Springer Verlag, Heidelberg Google Scholar
  4. Brouwer D. and Clemence G.M. (1961). Methods of Celestial Mechanics. Chapter XI. Academic Press, NY & Google Scholar
  5. Deprit A. (1969). Canonical transformations depending on a small parameter. Celest. Mech. 1: 12–30 MATHCrossRefADSGoogle Scholar
  6. Deprit A. and Elipe A. (1993). Complete reduction of the Euler-Poinsot problem. J Astronaut Sci 41(4): 603–628 Google Scholar
  7. Efroimsky, M.: Equations for the orbital elements. Hidden symmetry. Preprint no 1844 of the Institute of Mathematics and its Applications, University of Minnesota (2002a) preprints/feb02/feb02.htmlGoogle Scholar
  8. Efroimsky, M.: The implicit gauge symmetry emerging in the n-body problem of celestial mechanics (2002b) astro-ph/0212245Google Scholar
  9. Efroimsky M. and Goldreich P. (2003). Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach. J. Math. Phys. 44: 5958–5977. astro-ph/0305344MATHCrossRefADSGoogle Scholar
  10. Efroimsky M. and Goldreich P. (2004). Gauge freedom in the N-body problem of celestial mechanics. Astron. Astrophys. 415: 1187–1199. astro-ph/0307130CrossRefADSGoogle Scholar
  11. Efroimsky, M.: On the theory of canonical perturbations and its application to Earth rotation. Talk at the conference Journées 2004: Systèmes de référence spatio-temporels, l’Observatoire de Paris, 20–22 septembre (2004) astro-ph/0409282Google Scholar
  12. Efroimsky M. (2005). Long-term evolution of orbits about a precessing oblate planet. The case of uniform precession. Celest. Mech. Dynam. Astron. 91: 75–108. astro-ph/0408168MATHCrossRefADSGoogle Scholar
  13. Efroimsky M. (2006). Gauge freedom in orbital mechanics. Ann New York Acad Sci. 1065: 346–374. astro-ph/0603092CrossRefADSGoogle Scholar
  14. Escapa A., Getino J. and Ferrándiz J. (2001). Canonical approach to the free nutations of a three-layer Earth model. J. Geophys. Res. 106(B6): 11387–11397 CrossRefADSGoogle Scholar
  15. Escapa A., Getino J. and Ferrándiz J. (2002). Indirect effect of the triaxiality in the Hamiltonian theory for the rigid Earth nutations. Astron Astrophys. 389: 1047–1054 CrossRefADSGoogle Scholar
  16. Fukushima T. and Ishizaki H. (1994). Elements of spin motion. Celest. Mech. Dyn. Astron. 59: 149–159 MATHCrossRefADSGoogle Scholar
  17. Getino J. and Ferrándiz J. (1990). A Hamiltonian theory for an elastic earth. Canonical variables and kinetic energy. Celest. Mech. Dyn. Astron. 49: 303–326 CrossRefADSGoogle Scholar
  18. Getino J. and Ferrándiz J. (1994). A rigorous Hamiltonian approach to the rotation of elastic bodies. Celest. Mech. Dyn. Astron. 58: 277–295 MATHCrossRefADSGoogle Scholar
  19. Giacaglia G.E.O. and Jefferys W.H. (1971). Motion of a space station. I. Celest. Mech. 4: 442–467 Google Scholar
  20. Goldreich P. (1965). Inclination of satellite orbits about an oblate precessing planet. Astron. J. 70: 5–9 CrossRefADSGoogle Scholar
  21. Goldstein H. (1981). Classical Mechanics. Addison-Wesley, Reading MA Google Scholar
  22. Gurfil, P., Elipe, A., Tangren, W., Efroimsky, M.: The Serret-Andoyer formalism in rigit-body dynamics: I. Symmetries and perturbations. Submitted to Regular and Chaotic Dynamics (2007) astro-ph/0607201Google Scholar
  23. Hori G.-I. (1966). Theory of general perturbations with unspecified canonical variables. Publ. Astron. Soc. Jpn 18: 287–296 ADSGoogle Scholar
  24. Kholshevnikov, K.V.: Lie transformations in celestial mechanics. In: Astronomy and Geodesy. Thematic Collection of Papers, 4, Issue 4, pp. 21–45. Published by the Tomsk State University Press, Tomsk, Russia (1973) (in Russian)Google Scholar
  25. Kholshevnikov, K.V.: Asymptotic Methods of Celestial Mechanics, Chapter 5. Leningrad State University Press, St.Petersburg, Russia (1985) /in Russian/Google Scholar
  26. Kinoshita H. (1972). First-order perturbations of the two finite-body problem. Pub. Astron. Soc. Jpn. 24: 423–457 ADSGoogle Scholar
  27. Kinoshita H. (1977). Theory of the rotation of the rigid Earth. Celest. Mech. 15: 277–326 CrossRefADSGoogle Scholar
  28. Kinoshita H., Nakajima K., Kubo Y., Nakagawa I., Sasao T. and Yokoyama K. (1978). Note on nutation in ephemerides. Publi. Int. Latitude Observat. Mizusawa XII(1): 71–108 Google Scholar
  29. Kinoshita H. and Souchay J. (1990). The theory of the nutation for the rigid-Earth model at the second order. Celest. Mech. Dyn. Astron. 48: 187–265 CrossRefADSGoogle Scholar
  30. Laskar J. and Robutel J. (1993). The chaotic obliquity of the planets. Nature 361: 608–612 CrossRefADSGoogle Scholar
  31. Lieske J.H., Lederle T., Fricke W. and Morando B. (1977). Expressions for the precession quantities based upon the IAU /1976/ system of astronomical constants. Astron. Astrophys. 58: 1–16 ADSGoogle Scholar
  32. Mysen E. (2004). Rotational dynamics of subsolar sublimating triaxial comets. Planet. Space Sci. 52: 897–907 CrossRefADSGoogle Scholar
  33. Mysen E. (2006). Canonical rotation variables and non-Hamiltonian forces: solar radiation pressure effects on asteroid rotation. Monthly Notices Roy. Astron. Soc. 372: 1345–1350 CrossRefADSGoogle Scholar
  34. Peale S.J. (1973). Rotation of solid bodies in the solar system. Rev. Geophys. Space Phy. 11: 767–793 ADSGoogle Scholar
  35. Peale S.J. (1976). Excitation and relaxation of the wobble, precession and libration of the Moon. J Geophy. Res. 81: 1813–1827 ADSGoogle Scholar
  36. Petrov L. (2007). The empirical Earth-rotation model from VLBI observations. Astron Astrophys. 467: 359–369 CrossRefADSGoogle Scholar
  37. Plummer H.C. (1918). An Introductory Treatise on Dynamical Astronomy. Cambridge University Press, UK Google Scholar
  38. Poincaré, H. Sur une forme nouvelle des équations du problème des trois corps. Bull. Astron. 14, 53–67 (1897). For modern edition see: Œuvres de Henri Poincaré, Tome VII, pp. 500–511. Gauthier-Villars, Paris (1950)Google Scholar
  39. Radau, R.: Sur la rotation des corps solides. Annales de l’Ecole Normale Supérieure. 1resérie. Tome 6, 233–250 (1869) Scholar
  40. Richelot, F.J.: Eine neue Lœsung des Problemes der Rotation eines festen Körpers um einen Punkt. Abhandlungen der Königlichen Preuβischen Akademie der Wissenschaften zu Berlin. Math., 1–60 (1850)Google Scholar
  41. Seidelmann, P.K. (1992). Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley CAGoogle Scholar
  42. Serret J.A. (1866). Mémoire sur l’emploi de la méthode de la variation des arbitraires dans la théorie des mouvements de rotation. Mémoires de l’Academie des Sciences de Paris 55: 585–616 Google Scholar
  43. Schreiber K.U., Velikoseltsev A., Rothacher M., Klügel T., Stedman G.E. and Wiltshire D.L. (2004). Direct measurements of diurnal polar motion by ring laser gyroscopes. J. Geophys. Res. 109: B06405 CrossRefGoogle Scholar
  44. Souchay J., Losley B., Kinoshita H. and Folgueira M. (1999). Corrections and new developments in rigid Earth nutation theory. III. Final tables “REN-2000” including crossed-nutation and spin–orbit coupling effects. Astron. Astrophys. Suppl. 135, 111–131CrossRefADSGoogle Scholar
  45. Synge J.L. and Griffith B.A. (1959). Principles of Mechanics. McGraw-Hill, NY Google Scholar
  46. Tisserand F. (1889). Traité de mécanique Céleste. Gauthier-Villars, Paris Google Scholar
  47. Touma J. and Wisdom J. (1993). The chaotic obliquity of Mars. Science 259(5099): 1294–1297 CrossRefADSGoogle Scholar
  48. Touma J. and Wisdom J. (1994). Lie-Poisson integrators for rigid body dynamics in the solar system. Astron. J. 107: 1189–1202 CrossRefADSGoogle Scholar
  49. Zanardi M. and Vilhena de Moraes R. (1999). Analytical and semi-analytical analysis of an artificial satellite’s rotational motion. Celest. Mech. Dyn. Astron. 75: 227–250 MATHCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.US Naval ObservatoryWashingtonUSA
  2. 2.Departamento de Matemática AplicadaUniversidad de AlicanteAlicanteSpain

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