Celestial Mechanics and Dynamical Astronomy

, Volume 110, Issue 3, pp 189–198 | Cite as

The eleventh motion constant of the two-body problem

Original Article


The two-body problem is a twelfth-order time-invariant dynamic system, and therefore has eleven mutually-independent time-independent integrals, here referred to as motion constants. Some of these motion constants are related to the ten mutually-independent algebraic integrals of the n-body problem, whereas some are particular to the two-body problem. The problem can be decomposed into mass-center and relative-motion subsystems, each being sixth-order and each having five mutually-independent motion constants. This paper presents solutions for the eleventh motion constant, which relates the behavior of the two subsystems. The complete set of mutually-independent motion constants describes the shape of the state-space trajectories. The use of the eleventh motion constant is demonstrated in computing a solution to a two-point boundary-value problem.


Two-body problem Motion constants Boundary-value problems 


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  1. Battin R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, sect. 2.4 & 3.6 and chap. 4. AIAA, Virginia (1999)Google Scholar
  2. Bruns H.: Über die integrale des Vielkörper-problems. Acta Mathematica 11, 25–96 (1887)CrossRefMathSciNetGoogle Scholar
  3. Gronchi G.F., Dimare L., Milani A.: Orbit determination with the two-body integrals. Celest. Mech. Dyn. Astron. 107, 299–318 (2010)ADSCrossRefMathSciNetMATHGoogle Scholar
  4. Herrick S.: Universal variables. Astron. J. 70, 309–315 (1965)ADSCrossRefMathSciNetGoogle Scholar
  5. Isidori A.: Nonlinear Control Systems, 2 edn. sect. 1.4. Springer, Berlin (1989)Google Scholar
  6. Poincaré H.: Sur la méthode de Bruns. Comptes rendus hebdomadaires des séances de l’Académie des sciences 123, 1224–1228 (1896)Google Scholar
  7. Poincaré H.: Sur une forme nouvelle des équations de la mécanique. C.R. Acad. Sci. 132, 369–371 (1901)MATHGoogle Scholar
  8. Prussing J.E., Conway B.A.: Orbital Mechanics, sect. 1.2. Oxford University Press, New York, NY (1993)Google Scholar
  9. Roy A.E.: Orbital Motion, 4 edn. sect. 5.3. Taylor & Francis, New York (2005)Google Scholar
  10. Schaub H., Junkins J.L.: Analytical Mechanics of Space Systems, pp. 558. AIAA, Virgina (2003)Google Scholar
  11. Stumpff K.: Neue Formeln und Hilfstafeln zur Ephemeridenrechnung. Astron. Nachrichten 275, 108–128 (1947)ADSMATHCrossRefMathSciNetGoogle Scholar
  12. Szebehely V.: Theory of Orbits, sect. 1.3, 1.8 & 2.1. Academic Press, New York (1967)Google Scholar
  13. Vallado D.A.: Fundamentals of Astrodynamics and Applications, 2 edn. sect. 1.4.2. Microcosm Press, El Segundo, California (2001)Google Scholar
  14. Whittaker E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4 edn. sect. 37 & 155 and chap. 14. Cambridge University Press, Cambridge, UK (1970)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Aerospace Engineering DepartmentAuburn UniversityAuburnUSA
  2. 2.Aerospace Engineering DepartmentTexas A&M UniversityCollege StationUSA

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