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.

A Brief Account of G.D. Birkhoff’s Problem in the Problem of Three Bodies

Application of the theorem of connectivity of M7 in astronomy

The Analytical Foundations of Celestial Mechanics

Some topology of the 3-body problem

Some topology of n-body problems

https://ntrs.nasa.gov/api/citations/19720007912/downloads/19720007912.pdf

The topology of the regularized integral surfaces of the 3-body problem

Publisher Summary This chapter describes the qualitative aspects of the three-body flow. It describes a way in which the recent theory of isolating blocks may be applied to the Newtonian planar three-body problem. A full understanding of the three-body problem also requires application of the theory of quasi-periodic motion and perhaps a number of other as yet undiscovered ideas. Newton's model for the motion of three bodies in the plane under the influence of their mutual gravitational attraction consists of the manifold (R 6 – Δ) X R 6 of states of the system together with the equations of motion. The equations of motion form a Hamiltonian system of differential equations with Hamiltonian function H ( Q , P ). It is known that Newton's equations of motion conserve energy, momentum, and angular momentum. A first step in the qualitative study of the three-body problem might be to describe the topology of the manifolds of a surface M ( h, ω ). This has been done for the planar and the nonplanar problems. The chapter describes a general approach to the study of a flow suggested by Charles Conley. The approach follows the following steps: (1) locate certain isolated invariant sets of the flow; (2) describe the phase portrait locally near these sets; (3) establish how the invariant sets are tied together by heteroclinic and homoclinic orbits; and (4) describe the phase portrait near the union of the invariant sets and the heteroclinic and homoclinic orbits.

A Brief Account of G.D. Birkhoff’s Problem in the Problem of Three Bodies

Citations

Application of the theorem of connectivity of M7 in astronomy

References

The Analytical Foundations of Celestial Mechanics

Some topology of the 3-body problem

Some topology of n-body problems

The topology of the regularized integral surfaces of the 3-body problem

Some Qualitative Aspects of the Three-Body Flow*

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