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The group of an N-body problem in ce...

Chalmers, James Handson.

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The group of an N-body problem in celestial mechanics.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	The group of an N-body problem in celestial mechanics./
Author:	Chalmers, James Handson.
Description:	255 p.
Notes:	Source: Dissertation Abstracts International, Volume: 61-04, Section: B, page: 1978.
Contained By:	Dissertation Abstracts International61-04B.
Subject:	Applied Mechanics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=NQ48325
ISBN:	9780612483255

The group of an N-body problem in celestial mechanics.
Chalmers, James Handson.

The group of an N-body problem in celestial mechanics. - 255 p.

Source: Dissertation Abstracts International, Volume: 61-04, Section: B, page: 1978.

Thesis (Ph.D.)--Carleton University (Canada), 1999.

The N-body problem in celestial mechanics is studied for a force of attraction that is sufficiently general that initial value problems exist which do not exhibit uniqueness of solutions but are sufficiently limited that conservation of momentum, angular momentum and energy and invariance under orthogonal transformations are all obeyed. In this general setting and for arbitrary finite dimension of the space, a geometrical symmetry group is defined for a solution at any given time. Each member of the group permutes bodies of equal mass using related isometries in position and velocity space. It is shown that if the force of attraction is sufficiently pathological, the group can vary over time. However, if the attraction law yields uniqueness of solution, the group is preserved over time for the solutions. A family of coordinate frames, called centroidal is defined and is shown to characterize precisely those frames for which the position and velocity parts of the group reduce to identical orthogonal groups. It is seen that substantial reduction of the phase space can be accomplished for some problems using the group. Although others have studied many interesting examples of problems that involve such symmetry groups, the fact that each N-body problem has a group as shown herein has not been previously recognized.

ISBN: 9780612483255Subjects--Topical Terms:

1018410
Applied Mechanics.

The group of an N-body problem in celestial mechanics.
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020 $a 9780612483255
035 $a (UMI)AAINQ48325
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100 1 $a Chalmers, James Handson. $3 1675830
245 1 4 $a The group of an N-body problem in celestial mechanics.
300 $a 255 p.
500 $a Source: Dissertation Abstracts International, Volume: 61-04, Section: B, page: 1978.
500 $a Adviser: Angelo B. Mingarelli.
502 $a Thesis (Ph.D.)--Carleton University (Canada), 1999.
520 $a The N-body problem in celestial mechanics is studied for a force of attraction that is sufficiently general that initial value problems exist which do not exhibit uniqueness of solutions but are sufficiently limited that conservation of momentum, angular momentum and energy and invariance under orthogonal transformations are all obeyed. In this general setting and for arbitrary finite dimension of the space, a geometrical symmetry group is defined for a solution at any given time. Each member of the group permutes bodies of equal mass using related isometries in position and velocity space. It is shown that if the force of attraction is sufficiently pathological, the group can vary over time. However, if the attraction law yields uniqueness of solution, the group is preserved over time for the solutions. A family of coordinate frames, called centroidal is defined and is shown to characterize precisely those frames for which the position and velocity parts of the group reduce to identical orthogonal groups. It is seen that substantial reduction of the phase space can be accomplished for some problems using the group. Although others have studied many interesting examples of problems that involve such symmetry groups, the fact that each N-body problem has a group as shown herein has not been previously recognized.
520 $a For dimensions 2 and 3 the groups permitting non-zero angular momentum are found. In any finite dimensional space, if the force of attraction is strictly positive, the masses are positive and the group contains a hyperplane of reflection symmetry, then the solution leads either to or from a singularity or both. This result encompasses a large family of N-body problems. In dimensions 2 and 3 all groups containing this property are identified.
520 $a We next study the case of homogeneous forces of attraction to prove uniqueness theorems for central configurations (c.c.'s) for a whole family of symmetry groups in which the bodies form the vertices of concentric similar (homothetic) polygons, polyhedra and in general, polytopes. In these problems all bodies on any given polytope have the same mass but the masses of the bodies on different polytopes can be freely assigned independently of each other. The result is analogous to the Moulton-Euler theorem for collinear c.c.'s in that for each way of nesting the polytopes inside one another and for each way of assigning the (positive) masses for each polytope there is exactly one central configuration. Consequently, we obtain n! c.c.'s for each type of polytope where n is the number of polytopes in the nest. This result is obtained for nested equilateral triangles in R2, regular tetrahedra in R3 regular pentatopes in R4 and in general regular d + 1 vertex simplices in Rd. The same result is also obtained for nested squares in R2, octahedra in R3 and cross polytopes in Rd.
520 $a The group is also used to study non-homographic solutions a la Hill in three and four dimensions. Although these solutions cannot be written in the form of known functions, it is assured that neither collision nor non-collision singularities can occur. Furthermore, similar to Hill's study of the 3-body restricted problem, regions of space are determined which limit where the bodies can move based on the initial conditions.
590 $a School code: 0040.
650 4 $a Applied Mechanics. $3 1018410
650 4 $a Mathematics. $3 515831
650 4 $a Physics, Astronomy and Astrophysics. $3 1019521
690 $a 0346
690 $a 0405
690 $a 0606
710 2 $a Carleton University (Canada). $3 1018407
773 0 $t Dissertation Abstracts International $g 61-04B.
790 1 0 $a Mingarelli, Angelo B., $e advisor
790 $a 0040
791 $a Ph.D.
792 $a 1999
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=NQ48325