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The orbits of periodic solutions of many body problems. / Ahmed Eid Al-Saedi

Swansea University Author: Ahmed Eid Al-Saedi

Abstract

"This thesis is concerned with the equal mass many-body problem and the stability of periodic solutions, with Keplerian (Coulombic) potential and other potentials. The classical n-body problem is a system of ordinary differential equations that describes the motion of n particles moving under N...

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Published: 2001
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42792
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spelling 2018-08-02T16:24:30.4921985 v2 42792 2018-08-02 The orbits of periodic solutions of many body problems. bb01ea9e03d08ad10fc4201df41138e0 NULL Ahmed Eid Al-Saedi Ahmed Eid Al-Saedi true true 2018-08-02 "This thesis is concerned with the equal mass many-body problem and the stability of periodic solutions, with Keplerian (Coulombic) potential and other potentials. The classical n-body problem is a system of ordinary differential equations that describes the motion of n particles moving under Newton/Coulomb laws of motion, where the forces acting are the mutual gravitational attractions, Coulonibic interaction with the presence of a constant magnetic field. The main contributions of the present work are: 1. The e-equation which characterises the linear stability of the f-equation of the DTW-solutions and provides information about the stability of the non-planar periodic solutions of the many-body problems. To note that the method can be applied to any conservative Hamiltonian system with three degrees of freedom which can be reduced to two degrees of freedom using the Cylindrical polar coordinates (Chapter 3). 2. A numerical approach to solving the e-equation for our standard examples providing a set of illustrative systems that show that the general solution to the e-equation is well behaved. The comparison of numerical and approximate analytical solutions of the e-solution for the four body gravitational problem appears to be good. (Chapter 4). 3. New periodic solutions, weaving styles and chasing styles, with axial symetry and non-collison of the bodies, we describe the algebra and symmetry that allows us to reduce a full system of equations to just those for essentially one particle. Some of these styles provided the figure eight periodic solutions (Chapter 5). 4. We try to give approximate solutions for the new families of the weaving periodic solutions (Chapter 5). 5. DTW-periodic solutions with a Logarithmic potential energy. One interesting feature of these solutions is the appearance of double points in the initial data space corresponding to specified nodal structures. We also have the appearance of periodic orbits with the same nodal structure but different winding numbers. In the work of DTW these were denoted by use of a notation like "11/7", ie. 11 nodes with 7 revolutions required to complete the orbit (Chapter 5). 6. The extention of the use of the f-equation, the -equation and the numerical approach to other potentials (Chapter 5). 7. New style of periodic solution, the weaving style with the Logarithmic potential energy, this gave the figure eight periodic solution (Chapter 5). 8. Suggestion for further research areas in which one could continue this investigation (Chapter 5). (Abstract shortened by ProQuest.)." E-Thesis Mathematics. 31 12 2001 2001-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-02T16:24:30.4921985 2018-08-02T16:24:30.4921985 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Ahmed Eid Al-Saedi NULL 1 0042792-02082018162522.pdf 10807568.pdf 2018-08-02T16:25:22.2700000 Output 3284674 application/pdf E-Thesis true 2018-08-02T16:25:22.2700000 false
title The orbits of periodic solutions of many body problems.
spellingShingle The orbits of periodic solutions of many body problems.
Ahmed Eid Al-Saedi
title_short The orbits of periodic solutions of many body problems.
title_full The orbits of periodic solutions of many body problems.
title_fullStr The orbits of periodic solutions of many body problems.
title_full_unstemmed The orbits of periodic solutions of many body problems.
title_sort The orbits of periodic solutions of many body problems.
author_id_str_mv bb01ea9e03d08ad10fc4201df41138e0
author_id_fullname_str_mv bb01ea9e03d08ad10fc4201df41138e0_***_Ahmed Eid Al-Saedi
author Ahmed Eid Al-Saedi
author2 Ahmed Eid Al-Saedi
format E-Thesis
publishDate 2001
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description "This thesis is concerned with the equal mass many-body problem and the stability of periodic solutions, with Keplerian (Coulombic) potential and other potentials. The classical n-body problem is a system of ordinary differential equations that describes the motion of n particles moving under Newton/Coulomb laws of motion, where the forces acting are the mutual gravitational attractions, Coulonibic interaction with the presence of a constant magnetic field. The main contributions of the present work are: 1. The e-equation which characterises the linear stability of the f-equation of the DTW-solutions and provides information about the stability of the non-planar periodic solutions of the many-body problems. To note that the method can be applied to any conservative Hamiltonian system with three degrees of freedom which can be reduced to two degrees of freedom using the Cylindrical polar coordinates (Chapter 3). 2. A numerical approach to solving the e-equation for our standard examples providing a set of illustrative systems that show that the general solution to the e-equation is well behaved. The comparison of numerical and approximate analytical solutions of the e-solution for the four body gravitational problem appears to be good. (Chapter 4). 3. New periodic solutions, weaving styles and chasing styles, with axial symetry and non-collison of the bodies, we describe the algebra and symmetry that allows us to reduce a full system of equations to just those for essentially one particle. Some of these styles provided the figure eight periodic solutions (Chapter 5). 4. We try to give approximate solutions for the new families of the weaving periodic solutions (Chapter 5). 5. DTW-periodic solutions with a Logarithmic potential energy. One interesting feature of these solutions is the appearance of double points in the initial data space corresponding to specified nodal structures. We also have the appearance of periodic orbits with the same nodal structure but different winding numbers. In the work of DTW these were denoted by use of a notation like "11/7", ie. 11 nodes with 7 revolutions required to complete the orbit (Chapter 5). 6. The extention of the use of the f-equation, the -equation and the numerical approach to other potentials (Chapter 5). 7. New style of periodic solution, the weaving style with the Logarithmic potential energy, this gave the figure eight periodic solution (Chapter 5). 8. Suggestion for further research areas in which one could continue this investigation (Chapter 5). (Abstract shortened by ProQuest.)."
published_date 2001-12-31T03:53:39Z
_version_ 1763752667846279168
score 11.014067

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