Skip to main content
SpringerLink
Log in

On the Stability of Lagrange Solutions in the Spatial Near-Circular Restricted Three-Body Problem

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract—

The restricted problem of three bodies (material points) is considered. The orbits of the main attracting bodies are assumed to be ellipses of small eccentricity, and the passively gravitating body during its motion can leave the plane of the orbits of the main bodies (spatial problem). The stability of body motion corresponding to triangular Lagrangian libration points is investigated. A characteristic feature of the spatial problem under study is the presence of resonance due to the equality of the Keplerian motion period of the main bodies and the linear oscillation period of the passively gravitating body in the direction perpendicular to the plane of their orbits. Using the methods of classical perturbation theory, Kolmogorov—Arnold—Moser (KAM) theory and computer algebra algorithms, the nonlinear problem of stability for most (in the Lebesgue-measure sense) initial conditions and formal stability (stability in any arbitrarily high finite approximation with respect to the coordinates and impulses of perturbed motion) are investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. G. N. Duboshin, Celestial Mechanics: Analytical and Qualitative Methods (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  2. A. P. Markeev, Libration Points in Celestial Mechanics and Cosmodynamics (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  3. L. Euler, “De motu rectilineo trium corporum se mutuo attrahentum,” Novi Comm. Acad. Sci. Imp. Petrop. 11, 144–151 (1767).

    Google Scholar 

  4. J. L. Lagrange, Essai sur le Problème des Trois Corps. Oeuvres de Lagrange (Gauthier Villars, Paris, 1873), Vol. 6, pp. 229–324.

    Google Scholar 

  5. A. M. Lyapunov, “On the stability of motion in one particular case of the three-body problem,” in Collection of Works (Academy of Sciences of the USSR, Moscow, 1956), Vol. 1, pp. 327–401.

    Google Scholar 

  6. J. M. A. Danby, “Stability of the triangular points in the elliptic restricted problem of three bodies,” Astron. J. 69 (2), 165–172 (1964).

    Article  ADS  MathSciNet  Google Scholar 

  7. G. E. O. Giacaglia, “Characteristics exponents at L4 and L5 in the elliptic restricted problem of three bodies,” Celestial Mech. Dyn. Astron. 4 (3/4), 468–489 (1971).

    Article  ADS  MathSciNet  Google Scholar 

  8. A. H. Nayfeh and A. A. Kamel, “Stability of the triangular points in the elliptic restricted problem of three bodies,” AIAA J. 8 (2), 221–223 (1970).

    Article  ADS  Google Scholar 

  9. M. G. Yumagulov and O. N. Belikova, “Bifurcation of 4π-periodic solutions of the planar, restricted, elliptical three-body problem,” Astron. Rep. 86 (2), 148–152 (2009).

    Article  ADS  Google Scholar 

  10. T. Kovacs, “Stability chart of the triangular points in the elliptic restricted problem of three bodies,” Mon. Not. R. Astron. Soc. 430 (4), 2755–2760 (2013).

    Article  ADS  Google Scholar 

  11. N. R. Isanbaeva, “On the construction of the boundaries of stability regions of triangular libration points of a planar bounded elliptic three-body problem,” Vestn. Bashk. Univ., Mat. Mekh. 22 (1), 5–9 (2017).

    Google Scholar 

  12. C. Simo, “Periodic orbits of the planar N-body problem with equal masses and all bodies on the same path,” in Proc. 54th Scottish Universities Summer School in Physics “The Restless Universe. Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galactic Systems”, Blair Atholl, July 23–Aug. 5, 2000 (CRC Press, New York, 2001), pp. 265–284.

  13. A. L. Whipple and V. Szebehely, “The restricted problem of n + v bodies,” Celestial Mech. Dyn. Astron. 32 (2), 137–144 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  14. D. A. Budzko and A. N. Prokopenya, “On the stability of equilibrium positions in the circular restricted four-body problem,” in Computer Algebra in Scientific Computing, Ed. by V. P. Gerdt (Springer-Verlag, Heidelberg, 2011), Vol. 6885, pp. 88–100.

    MATH  Google Scholar 

  15. E. A. Grebenikov, Mathematical Problems of Homographic Dynamics (Peoples’ Friendship University of Russia, Moscow, 2011).

    Google Scholar 

  16. A. L. Kunitsyn and A. T. Turesbaev, “Stability of triangular libration points of the photogravitational three-body problem,” Pis’ma Astron. Zh. 11 (2), 145–148 (1985).

    ADS  Google Scholar 

  17. L. G. Luk’yanov and A. Yu. Kochetkova, “On the stability of Lagrangian libration points in a restricted elliptic photogravitational three-body problem,” Vestn. Mosk. Univ., Ser. 3: Fiz., Astron., No. 5, 71–76 (1996).

  18. A. S. Zimovshikov and V. N. Tkhai, “Stability diagrams for a heterogeneous ensemble of particles at the collinear libration points of the photogravitational three-body problem,” J. Appl. Math. Mech. 74 (2), 158–163 (2010).

    Article  MathSciNet  Google Scholar 

  19. A. Kononenko, “Libration points of the Earth–Moon system,” Aviats. Kosmonavt., No. 5, 71–73 (1968).

  20. N. F. Averkiev, S. A. Vas’kov, and V. V. Salov, “Ballistic construction of communication spacecraft systems and passive radar of the Lunar surface,” Izv. Vyssh. Uchebn. Zaved., Priborostr. 51 (12), 66–72 (2008).

    Google Scholar 

  21. F. Salazar, O. Winter, E. Macau, J. Masdemont, and G. Gomez, “Natural configuration for formation flying around triangular libration points for the elliptic and the bicircular problem in the Earth–Moon system,” in Proc. 65th Int. Astronautical Congress (Toronto, 2014).

  22. I. G. Malkin, Theory of Stability of Motion (Office Techn. Inform, Washington, 1952).

    MATH  Google Scholar 

  23. A. P. Markeev, “On normal voordinates in the vicinity of the Lagrangian libration points of the restricted elliptic three-body problem,” Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 30 (4), 657–671 (2020).

    Article  Google Scholar 

  24. G. E. O. Giacaglia, Perturbation Methods in Non-Linear Systems (Springer, New York, 1972).

    Book  Google Scholar 

  25. A. X. Nayfeh, Perturbation Methods (Wiley, New York, 1973).

    MATH  Google Scholar 

  26. A. N. Kolmogorov, “Preservation of conditionally periodic movements with small change in the Hamilton function,” in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems, Ed. by G. Casati and J. Ford (Springer-Verlag, Berlin, 1979), Vol. 93, pp. 51–56.

    Google Scholar 

  27. V. I. Arnol’d, “Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian,” Russ. Math. Surv. 18 (5), 9–36 (1963).

    Article  Google Scholar 

  28. V. I. Arnol’d, “Small denominators and problems of stability of motion in classical and celestial mechanics,” Russ. Math. Surv. 18 (6), 85–191 (1963).

    Article  MathSciNet  Google Scholar 

  29. V. I. Arnol’d, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).

    Book  Google Scholar 

  30. V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics Encyclopaedia Math. Sci. Ser., vol. 3 (Springer-Verlag, Berlin, 2006).

    Book  Google Scholar 

  31. Ch. Lhotka, C. Efthymiopoulos, and R. Dvorak, “Nekhoroshev stability at L4 or L5 in the elliptic-restricted three-body problem-application to Trojan asteroids,” Mon. Not. R. Astron. Soc. 384, 1165–1177 (2008).

    Article  ADS  Google Scholar 

  32. A. Deprit and A. Deprit-Bartholomé, “Stability of the triangular Lagrangian points,” Astron. J. 72 (2), 173–179 (1967).

    Article  ADS  Google Scholar 

  33. A. P. Markeev, “On the stability of triangular Lagrangian solutions in the spatial circular restricted three-body problem,” Astron. Zh 48 (4), 862–868 (1971).

    ADS  MATH  Google Scholar 

  34. N. N. Nekhoroshev, “Behavior of Hamiltonian systems close to integrable,” Funct. Anal. Appl. 5 (4), 338–339 (1971).

    Article  Google Scholar 

  35. N. N. Nekhoroshev, “An exponential estimate of the stability time of near-integrable Hamiltonian systems,” Russ. Math. Surv. 32 (6), 1–65 (1977).

    Article  Google Scholar 

  36. N. N. Nekhoroshev, “An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II,” in Topics in Modern Mathematics. Petrovskii Seminar, Ed. by O. A. Oleinik (Consultant Bureau, New York, 1985), No. 5, pp. 1–58.

  37. J. Moser, “New aspects in the theory of stability of Hamiltonian systems,” Commun. Pure Appl. Math. 11 (1), 81–114 (1958).

    Article  MathSciNet  Google Scholar 

  38. J. Moser, “Stabilitatsverhalten Kanonischer Differentialgleichungs Systeme,” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa 1955, 87–120 (1955).

    Google Scholar 

  39. J. Moser, “On the elimination of the irrationality condition and Birkhoff'’s concept of complete stability,” Bol. Soc. Mat. Mexicana 5, 167–175 (1960).

    MathSciNet  MATH  Google Scholar 

  40. J. Moser, “Stability of the asteroids,” Astron. J. 63 (10), 439–443 (1958).

    Article  ADS  Google Scholar 

  41. C. L. Siegel, Vorlesungen über Himmelsmechanik, Grundlehren Math. Wiss., vol. 85 (Springer-Verlag, Berlin, 1956)).

  42. J. Glimm, “Formal stability of Hamiltonian systems,” Commun. Pure Appl. Math. 17 (4), 509–526 (1964).

    Article  MathSciNet  Google Scholar 

  43. A. D. Bruno, “Formal stability of Hamiltonian systems,” Math. Zametki 1 (3), 216–219 (1967).

    MathSciNet  Google Scholar 

  44. A. D. Bruno, The Restricted 3-Body Problem: Plane Periodic Orbits (Walter de Gruyter, Berlin, 1994), Vol. 17.

    Book  Google Scholar 

Download references

Funding

This work was supported by a grant from the Russian Science Foundation (project no. 19-11-00116) at Moscow Aviation Institute (National Research University) and at Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    A. P. Markeev

  2. Moscow Aviation Institute, 125993, Moscow, Russia

    A. P. Markeev

Authors
  1. A. P. Markeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. P. Markeev.

Ethics declarations

The author declares that he has no conflict of interest.

Additional information

Translated by T. N. Sokolova

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Markeev, A.P. On the Stability of Lagrange Solutions in the Spatial Near-Circular Restricted Three-Body Problem. Mech. Solids 56, 1541–1549 (2021). https://doi.org/10.3103/S0025654421080124

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654421080124

Keywords:

Search

Navigation