On the Lagrange Stability of Motion and Final Evolutions in the Three-Body Problem

Stepan P. Sosnitskii

Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine.

DOI: 10.4236/am.2013.42057                

Abstract

For the three-body problem, we consider the Lagrange stability. To analyze the stability, along with integrals of energy and angular momentum, we use relations by the author from [1], which band together separately squared mutual distances between bodies (mass points) and squared distances from bodies to the barycenter of the system. In this case, we prove the Lagrange stability theorem, which allows us to define more exactly the character of hyperbolic-elliptic and parabolic-elliptic final evolutions.

Keywords

Lagrange Stability; Distal Motion; Hill Stable Pair; Final Evolutions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. P. Sosnitskii, “On the Lagrange Stability of the Motion for the Three-Body Problem,” Ukrainian Mathematical Journal, Vol. 57, No. 8, 2005, pp. 1341-1349.
[2]	E. T. Whittaker, “A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,” Dover, New York, 1944.
[3]	G. N. Duboshin, “Celestial Mechanics. Analitical and Qualitative Methods,” Nauka, Moscow, 1964.
[4]	L. A. Pars, “A Treatise on Analytical Dynamics,” Heinemann, London, 1965.
[5]	V. I. Arnold, “Proof of a Theorem by A. N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian,” Uspekhi Matematicheskikh Nauk, Vol. 18, No. 5, 1963, pp. 13-40.
[6]	S. P. Sosnitskii, “On the Orbital Stability of Triangular Lagrangian Motions in the Three-Body Problem,” Astronomical Journal, Vol. 136, No. 6, 2008, pp. 2533-2540. doi:10.1088/0004-6256/136/6/2533
[7]	A. E. Roy, “Orbital Motion,” Adam Hilger LTD, Bristol, 1978.
[8]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, “Mathematical Aspects of Classical and Celestial Mechanics,” URSS, Moscow, 2002.
[9]	V. G. Golubev and E. A. Grebenikov, “The Three-Body Problem in the Celestial Mechanics,” Moscow Univ. Publ., Moscow, 1985.
[10]	C. Marchal, “The Three-Body Problem,” Elsevier, Oxford, 1990.
[11]	L. G. Luk’yanov and G. I. Shirmin, “Sundman Surfaces and Hill Stability in the Three-Body Problem,” Letters to the Astronomical Journal, Vol. 33, No. 8, 2007, pp. 618-630.
[12]	S. P. Sosnitskii, “On the Lagrange and Hill Stability of the Motion of Certain Systems with Newtonian Potential,” Astronomical Journal, Vol. 117, No. 6, 1999, pp. 3054-3058. doi:10.1086/300889
[13]	J. Chazy, “Sur l’Allure Finale Mouvement dans le Probleme des Trois Corps Quand le Temps Croit Indefiniment,” Annales de l’Ecole Normale Superieure, 3eme Serie, Vol. 39, 1922, pp. 29-130.