Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T04:03:22.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrals of Motion

Published online by Cambridge University Press:  14 August 2015

G. Contopoulos
G. Contopoulos*
Affiliation:
University of Maryland, U.S.A. University of Thessaloniki, Greece

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The properties of conservative dynamical systems of two or more degrees of freedom are reviewed. The transition from integrable to ergodic systems is described. Nonintegrability is due to the interaction of two, or more, resonances. Then one sees, on a surface of section, infinite types of islands of various orders, while the asymptotic curves from unstable invariant points intersect each other along homoclinic and heteroclinic points producing an apparent ‘dissolution’ of the invariant curves. A threshold energy is defined separating near integrable systems from near ergodic ones. The possibility of real ergodicity for large enough energies is discussed. In the case of many degrees of freedom we also distinguish between integrable, ergodic, and intermediate cases. Among the latter are systems of particles interacting with Lennard-Jones interparticle potential. A threshold energy was derived, which is proportional to the number of particles. Finally some recent results about the general three-body problem are described. One can extend the families of periodic orbits of the restricted problem to the general three-body problem. Many of these orbits are stable. An empirical study of orbits near the stable periodic orbits indicates the existence of 2 integrals of motion besides the energy.

Type
Part II/Flattened Systems
Information
Symposium - International Astronomical Union , Volume 69: Dynamics of Stellar Systems , 1975 , pp. 209 - 225
Copyright
Copyright © Reidel 1975 

References

Arnold, V. I.: 1963, Uspekhi Mat. Nauk 18, 91; Transl. Air Force Systems Command, Wright Patterson Air Force Base, Ohio.Google Scholar
Arnold, V. I. and Avez, A.: 1967, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris.Google Scholar
Birkhoff, G. D.: 1927, Dynamical Systems, American Math. Soc., Providence, R.I. Google Scholar
Chirikov, B. V.: 1971, Nucl. Phys. Inst. Siberian Sect. USSR Acad. Sci., Rep. No. 267; CERN Transl. 71–40.Google Scholar
Contopoulos, G.: 1960, Z. Astrophys. 49, 273.Google Scholar
Contopoulos, G.: 1966, Astrophys. J. Suppl. 13, 503.Google Scholar
Contopoulos, G.: 1967, Bull. Astron. Ser. 3, 2, 223.Google Scholar
Contopoulos, G.: 1970, Astron. J. 75, 108.Google Scholar
Contopoulos, G.: 1973, in Martinet, L. and Mayor, M. (eds.), Dynamical Structure and Evolution of Stellar Systems, Swiss Society of Astronomy and Astrophysics Third Advanced Course, Geneva Observatory, p. 52.Google Scholar
Einstein, A. and Stern, O.: 1913, Ann. Phys. 40, 551.Google Scholar
Fermi, E., Pasta, J., and Ulam, S.: 1955, Los Alamos Sci. Lab. Rep. LA-1940.Google Scholar
Ford, J.: 1973, Adv. Chem. Phys. 24, 155.Google Scholar
Galgani, L. and Scotti, A.: 1972, Rev. Nuovo Cimento 2, 189.Google Scholar
Hénon, M.: 1974a, Phys. Rev. Letters (in press).Google Scholar
Hénon, M.: 1974b, Celes. Mech. 10, 375.CrossRefGoogle Scholar
Hénon, M. and Heiles, C.: 1964, Astron. J. 69, 73.Google Scholar
Martinet, L.: 1974, Astron. Astrophys. 32, 329.Google Scholar
Miura, R. M., Gardner, C. S., and Kruskal, M. D.: 1968, J. Math. Phys. 9, 1204.Google Scholar
Moser, J.: 1962, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl., 1.Google Scholar
Moser, J.: 1973, Stable and Random Motions in Dynamical Systems, Princeton Univ. Press.Google Scholar
Poincaré, H.: 1899, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris.Google Scholar
Sinai, Ya.: 1970, Russian Math. Surveys 25, 137.Google Scholar
Stoddard, S. D. and Ford, J.: 1973, Phys. Rev. A 8, 1504.CrossRefGoogle Scholar
Walker, G. H. and Ford, J.: 1969, Phys. Rev. 188, 416.Google Scholar
Whittaker, E. T.: 1904, Analytical Dynamics of Particles and Rigid Bodies, Cambridge Univ. Press.Google Scholar
Zabusky, N. J.: 1967, in Nonlinear Partial Differential Equations, Academic Press, New York.Google Scholar
Zakharov, V. E. and Faddeev, L. D.: 1972, Funct. Anal. and Appl. 5, 280.Google Scholar