Physics
Scientific paper
Jan 1993
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993phdt........29t&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF MINNESOTA, 1993.Source: Dissertation Abstracts International, Volume: 54-09, Section: B, page: 471
Physics
3
Scientific paper
This paper analyzes central configurations which are special configurations leading to homothetic solutions of the n-body problem. For the planar central configurations, these solutions also provide periodic solutions of the n-body problem. Chapter 1 defines the problem and provides an overview of the area. There is historical interest in knowing the total number of these central configurations. For n >= 4 the problem remains unsolved. Furthermore, a different mass ratio between the n bodies will produce a different total number of central configurations. This paper will provide the total number of central configurations which has a special mass ratio. For the planar central configurations the total number of central configurations grows at the speed n!2n, and the three dimensional case is n!3^{n }. Chapters 2, 3 and 5 give details and proofs of the analytical continuation method. This method begins with three bodies, then creates the central configurations with four bodies with one small mass, using the implicit function theorem. If the process is repeated, the total number of central configurations for any n-body problem may be calculated, provided (n-3) masses are sufficiently small. In Chapters 4 and 7, the formulas are derived for the total number of central configurations of the n-body problem with special mass ratio (m _1,m_2,m_3,epsilon_1, ...,epsilonn) in both planar and three-dimensional cases. Examples of formulas provided are: n!(2^{n+1} + 1) , n!((n^2 - n + 4)2^{n+1 } - n - 7), and (n!/6) ((n ^3 + 11n - 12)2^{n+2} + 6n + 54). Chapter 6 solves a very special degenerate case during the continuation process. The Morse Index of these central configurations is discussed in Chapter 8.
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