Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

From the restricted to the full three-body problem


Authors: Kenneth R. Meyer and Dieter S. Schmidt
Journal: Trans. Amer. Math. Soc. 352 (2000), 2283-2299
MSC (2000): Primary 70F05, 37N05
DOI: https://doi.org/10.1090/S0002-9947-00-02542-3
Published electronically: February 16, 2000
MathSciNet review: 1694376
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.


References [Enhancements On Off] (What's this?)

  • 1. R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin Cummings, London, 1978. MR 81e:58025
  • 2. V. I. Arnold (Ed.), Encyclopaedia of Mathematical Sciences, Dynamical Systems III, Springer-Verlag, 1993. MR 95d:58043a
  • 3. E. A. Belbruno, A new family of periodic orbits for the restricted problem, Celestial Mech., 25, 1981, 397-415. MR 82m:70017
  • 4. G. D. Birkhoff, The restricted problem of three-bodies, Rend. Circolo Mat. Palermo, 39, 1915, 255-334.
  • 5. -, Dynamical Systems, Amer. Math. Soc., Providence, RI, 1927.
  • 6. D. Buchanan, Trojan satellites -- limiting case, Trans. Roy. Soc. Canada Sect. III (3), 35, 1941, 9-25. MR 3:216a
  • 7. C. C. Conley, Some new long period solutions of the plane restricted body problem of three-bodies, Comm. Pure Appl. Math., 16, 1963, 449-467. MR 27:4670
  • 8. A. Deprit and A. Deprit-Bartholomé, Stability of the Lagrange points, Astron. J., 72, 1967, 173-179.
  • 9. A. Deprit and J. Henrard, A manifold of periodic solutions, Advan. Astron. Astrophy, 6, 1968, 6-12.
  • 10. M. Golubitsky, J. Marsden, I Stewart, and Michael Dellnitz, The constrained Lyapunov-Schmidt procedure and periodic orbits, Fields Institute Communications, Vol. 4, 81-127, 1995. MR 96g:58135
  • 11. J. D. Hadjidemetriou, The continuation of periodic orbits from the restricted to the general three-body problem, Celestial Mech., 12, 1975, 155-174. MR 52:7243
  • 12. A. Liapounoff [Lyapunov], Problème générale de la stabilité du mouvement, Ann. Math. Studies 17, Princeton Univ. Press, Princeton, NJ, 1947. (Reproduction of the French translation.) MR 9:34j
  • 13. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetries, Rep. Math. Phy., 5, 1974, 121-130. MR 53:6633
  • 14. K. R. Meyer, Comet like periodic orbits in the $N$-body problem, J. Comp. and Appl. Math., 52, 1994, 337-351. MR 95k:70026
  • 15. -, Generic bifurcation of periodic points, Trans. Amer. Math. Soc., 149, 1970, 95-107. MR 41:3931
  • 16. -, Generic stability properties of periodic points, Trans. Amer. Math. Soc., 154, 1971, 273-277. MR 42:6373
  • 17. -, Periodic orbits near infinity in the restricted $N$-body problem, Celest. Mech., 23, 1981, 69-81. MR 82c:70015
  • 18. -, Periodic solutions of the $N$-body problem, J. Dif. Eqs., 39(1), 1981, 2-38. MR 83b:70016
  • 19. -, Symmetries and integrals in mechanics, Dynamical Systems (M. Peixoto, Ed.), Academic Press, New York, 1973, 259-272. MR 48:9760
  • 20. K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer-Verlag, New York, 1992. MR 93b:70002
  • 21. K. R. Meyer and J. I. Palmore, A new class of periodic solutions in the restricted three-body problem, J. Diff. Eqs., 44, 1970, 263-272. MR 46:7640
  • 22. K. R. Meyer and D. S. Schmidt, Periodic orbits near $\mathcal{L}_4$ for mass ratios near the critical mass ratio of Routh, Celest. Mech., 4, 1971, 99-109. MR 47:6164
  • 23. F. R. Moulton, A class of periodic solutions of the problem of three-bodies with applications to lunar theory, Trans. Amer. Math. Soc., 7, 1906, 537-577.
  • 24. -, A class of periodic orbits of the superior planets, Trans. Amer. Math. Soc., 13, 1912, 96-108.
  • 25. J. I. Palmore, Bridges and Natural Centers in the Restricted Three-Body Problem, University of Minnesota Report, Minneapolis MN, 1969. MR 43:8279
  • 26. O. Perron, Neue periodische Lösungen des ebenen Drei und Mehrkörperproblem, Math. Z., 42, 1937, 593-624.
  • 27. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauther-Villars, Paris, 1899. MR 89e:01054 (reprint)
  • 28. D. S. Schmidt, Periodic solutions near a resonant equilibrium of a Hamiltonian system, Celest. Mech., 9, 1974, 81-103. MR 49:9340
  • 29. -, Transformation to versal normal form, Computer Aided Proofs in Analysis, (Ed. K. R. Meyer and D. S. Schmidt), IMA Series 28, Springer-Verlag, 1990, 235-240. MR 92b:58210
  • 30. -, Versal normal form of the Hamiltonian function of the restricted problem of three-bodies near $\mathcal{L}_4$, J. Computational and Appl. Math., 52, 1994, 155-176. MR 95k:70020
  • 31. C. L. Siegel, Über eine periodische Lösung im Dreikörperproblem, Math. Nachr. , 4, 1950, 28-64. MR 14:910d
  • 32. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971. MR 58:19464
  • 33. A. Sokol$'$skii, On the stability of an autonomous Hamiltonian system, J. Appl. Math. Mech., 38, 1978, 741-49. MR 57:10734
  • 34. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, 1927.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 70F05, 37N05

Retrieve articles in all journals with MSC (2000): 70F05, 37N05


Additional Information

Kenneth R. Meyer
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221–0025
Email: ken.meyer@uc.edu

Dieter S. Schmidt
Affiliation: Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, Ohio 45221–0030
Email: dieter.schmidt@uc.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02542-3
Keywords: Restricted three--body problem, three--body problem, reduction, symplectic scaling, normal forms, KAM theory
Received by editor(s): July 22, 1997
Received by editor(s) in revised form: January 20, 1998
Published electronically: February 16, 2000
Additional Notes: This research was partially supported by grants from the National Science Foundation and the Taft Foundation.
Dedicated: To Hugh Turrittin on his ninety birthday
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society