Skip to main content
Log in

A brief account of G.D. Birkhoff's problem in the problem of three bodies

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The present article gives a historical survery of G.D. Birkhoff's seventh problem which is an inquiry about the topological structure of the set of definition of the reduced differential equations of motion. Recent advances in the problem and their meaning have been briefly indicated.

The classical 3-body problem concerns how the three particles should move under their mutual Newtonian attraction. By a particle we mean a goometrical point endorsed with a constant positive numberm which is called mass. Expressed mathematically, the problem appears as to solving of the following system of differential equations:

$$\begin{gathered} \frac{{dq_i }}{{dt}} = \dot q_i m_i \frac{{d\dot q_i }}{{dt}} = \frac{{\partial U}}{{\partial \dot q_i }}(q = x,y,z,i = 1,2,3) \hfill \\ U = \sum\limits_{i \ne k} {\frac{{m_i m_k }}{{r_{ik} }}} \hfill \\ \end{gathered} $$
((1))

wherex i,y i,z i signify the rectangular coordinates of thei-th particle andr ij denotes the distance between thei -th and the j-th particle. There are ten algebraic integrals of the system (1) describing the motion of the centre of gravity of the particles togather with conservation of angular momentum and conservation of energy.

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

Access this article

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. Birkhoff, G. D. Einige Problem der Dynamik, Jahresbericht der Deutschen Mathematikers Vereinigung (1929)38, Abt. Heft 1/4, 1–16. Collected Works 2 778–793.

  2. Birkhoff, G. D., Dynamical Systems, (1922) American Mathemaitcal Society Colloquium Publications.

  3. Kolmogrof, The general theory of dynamical systems and classical mechanics, (1954) International Congress of Mathematics. Amsterdam.

  4. Wintner, A., The analytical foundations of celestial mechanics.

  5. Easton, R. W., Some topology of the 3-body problem,Journal of Differential Equations, 10, 2 (1971)

    Google Scholar 

  6. Easton, R. W., Some topology of n-body problems,Journal of Differential Equations, 19, 2 (1975)

    Google Scholar 

  7. Easton, R. W., The topology of the regularited integral surfaces of the 3-body problem, Jour. of D. E. 12, 2 (1972)

    Google Scholar 

  8. Easton, R. W., Some qualitative aspects of the 3-body flow, Dynamical System, No 2, An international symposium (1976).

  9. Tung chin-chu, Some properties of the classical integrals of the general 3-body problem, Scientia Sinica, Vol. 17, No. 3 (1974)

  10. Tung, Chin-chu, The motion of Triton as a satellite of Neptune, Publications of the Beijing Astronomical Observatory, No. 15 (1978), (in Chinese)

  11. Tung, Chiu-chu, A property of the motion of the moon, Publications of the Beijing Astronomical Observatory, No. 1 (1979) (in Chinese).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chin-Chu, T. A brief account of G.D. Birkhoff's problem in the problem of three bodies. Appl Math Mech 1, 225–230 (1980). https://doi.org/10.1007/BF01876746

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01876746

Keywords

Navigation