Collisions, rings, and other Newtonian N-body problems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2005
|
| Series: | CBMS regional conference series in mathematics
104 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 223-231) and index |
| Physical Description: | x, 235 p. Ill., graph. Darst. 26 cm |
| ISBN: | 0821832506 |
Staff View
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| 245 | 1 | 0 | |a Collisions, rings, and other Newtonian N-body problems |c Donald G. Saari |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2005 | |
| 300 | |a x, 235 p. |b Ill., graph. Darst. |c 26 cm | ||
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| 490 | 1 | |a CBMS regional conference series in mathematics |v 104 | |
| 500 | |a Includes bibliographical references (p. 223-231) and index | ||
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Record in the Search Index
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| adam_text | Contents
Preface v
1 Introduction 1
1.1 Mars 1
1.1.1 Motion of Mars 2
1.1.2 The far out planets 4
1.2 Mercury 6
1.3 Epicycles 11
1.4 Chaotic behavior 13
1.4.1 Newton s method 14
1.4.2 Period three and circle maps 19
1.4.3 The forced Van der Pol equations 23
1.5 The rings of Saturn 26
1.5.1 Kinky behavior 26
1.5.2 A model 27
2 Central configurations 31
2.1 Equations of motion and integrals 32
2.2 Central Configurations 34
2.2.1 Why central configurations are important 36
2.2.2 Value of A 40
2.2.3 Equivalence classes of configurations 42
2.3 A conjecture and a velocity decomposition 47
2.3.1 Virial Theorem and the conjecture 47
2.3.2 The system velocity decomposition 51
2.3.3 Central configurations and the velocity decomposition 54
2.3.4 Motion preserving an Euler similarity class 56
vii
viii CONTENTS
2.3.5 Sundman inequality 61
2.4 More conjectures 65
2.4.1 Another conjecture 65
2.4.2 Special cases 67
2.5 Jacobi coordinates help see the dynamics 69
2.5.1 Velocity decomposition and a basis 71
2.5.2 Describing p with the basis 74
2.5.3 Seeing the gradient of U 76
2.5.4 An illustrating example 77
2.5.5 Finding central and other configurations . 79
2.5.6 Equations of motion for constant / 80
2.5.7 Basis for the coplanar iV body problem 81
3 Finding Central Configurations 83
3.1 From the ancient Greeks to 84
3.1.1 Arithmetic and geometric means 84
3.1.2 Connection with central configurations 88
3.2 Constraints 92
3.2.1 Singularity structure of F 94
3.2.2 Some dynamics 97
3.2.3 Stratified structure of the image of F 99
3.3 Geometric approach—the rule of signs 102
3.3.1 The configurational averaged length iCAL 103
3.3.2 Signs of gradients coplanar configurations 105
3.3.3 Signs of gradients three dimensional configurations . . 106
3.3.4 Degenerate configurations 107
3.4 Consequences for central configurations 109
3.4.1 Surprising regularity 109
3.4.2 Estimates on £CAL 112
3.4.3 Are there central configurations of these types? .... 114
3.5 What can, and cannot, be 115
3.5.1 More central configurations 115
3.5.2 Masses and collinear central configurations 119
3.5.3 Masses and coplanar configurations 124
3.6 New kinds of constraints 126
3.7 Rings of Saturn .... 130
3.7.1 Stability 131
3.7.2 More rings 133
3.7.3 Saturn, and some problems 136
CONTENTS ix
4 Collisions both real and imaginary 137
4.1 One body problem 138
4.1.1 Levi Civita s approach 140
4.1.2 Kustaanheimo and Stiefel s approach 141
4.1.3 Topological obstructions and hairy balls 143
4.1.4 Sundman s solution of the three body problem .... 144
4.2 Sundman and the three body problem 147
4.2.1 Complex singularities? 147
4.2.2 Avoiding complex singularities 148
4.2.3 Singularities retaliate 149
4.3 Generalized Weierstrass Sundman theorem 150
4.3.1 A simple case the central force problem 151
4.3.2 Larger p values and Black Holes 151
4.3.3 Lagrange Jacobi equation 153
4.3.4 Proof of the Weierstrass Sundman Theorem 154
4.3.5 Bounded above means bounded below 159
4.3.6 Problems 161
4.3.7 An interesting historical footnote 161
4.4 Singularities an overview 162
4.4.1 Behavior of a singularity 163
4.4.2 Non collision singularities 165
4.4.3 Off to infinity 166
4.4.4 Problems 170
4.5 Rate of approach of collisions 172
4.5.1 General collisions 172
4.5.2 Tauberian Theorems 173
4.5.3 Proof of the theorem 179
4.6 Sharper asymptotic results 184
4.7 Spin, or no spin? 185
4.7.1 Using the angular momentum 187
4.7.2 The collinear case 189
4.8 Manifolds defined by collisions 191
4.8.1 Structure of collision sets 192
4.8.2 Proof of Theorem 4.18 193
4.9 Proof of the slowly varying assertion 195
4.9.1 Centers of mass 198
4.9.2 Back to the proof 201
4.9.3 The last steps 203
x CONTENTS
5 How likely is it? 207
5.1 Motivation 208
5.1.1 Idea of proof 209
5.1.2 Why do we need the Baire category statement? .... 210
5.2 Proof: C is of first Baire category 210
5.2.1 Finding an appropriate C subset 211
5.2.2 A comment about the set of singularities 213
5.3 Proof: C is of Lebesgue measure zero 214
5.3.1 A common collision for k 3 particles 214
5.3.2 Lower dimensions, binary collisions, and other force
laws 217
5.4 Likelihood of non collision singularities 220
Bibliography 223
Index 232
|
| adam_txt |
Contents
Preface v
1 Introduction 1
1.1 Mars 1
1.1.1 Motion of Mars 2
1.1.2 The "far out" planets 4
1.2 Mercury 6
1.3 Epicycles 11
1.4 Chaotic behavior 13
1.4.1 Newton's method 14
1.4.2 Period three and circle maps 19
1.4.3 The forced Van der Pol equations 23
1.5 The rings of Saturn 26
1.5.1 Kinky behavior 26
1.5.2 A model 27
2 Central configurations 31
2.1 Equations of motion and integrals 32
2.2 Central Configurations 34
2.2.1 Why central configurations are important 36
2.2.2 Value of A 40
2.2.3 Equivalence classes of configurations 42
2.3 A conjecture and a velocity decomposition 47
2.3.1 Virial Theorem and the conjecture 47
2.3.2 The system velocity decomposition 51
2.3.3 Central configurations and the velocity decomposition 54
2.3.4 Motion preserving an Euler similarity class 56
vii
viii CONTENTS
2.3.5 Sundman inequality 61
2.4 More conjectures 65
2.4.1 Another conjecture 65
2.4.2 Special cases 67
2.5 Jacobi coordinates help "see" the dynamics 69
2.5.1 Velocity decomposition and a basis 71
2.5.2 Describing p" with the basis 74
2.5.3 "Seeing" the gradient of U 76
2.5.4 An illustrating example 77
2.5.5 Finding central and other configurations . 79
2.5.6 Equations of motion for constant / 80
2.5.7 Basis for the coplanar iV body problem 81
3 Finding Central Configurations 83
3.1 From the ancient Greeks to 84
3.1.1 Arithmetic and geometric means 84
3.1.2 Connection with central configurations 88
3.2 Constraints 92
3.2.1 Singularity structure of F 94
3.2.2 Some dynamics 97
3.2.3 Stratified structure of the image of F 99
3.3 Geometric approach—the rule of signs 102
3.3.1 The "configurational averaged length" iCAL 103
3.3.2 Signs of gradients coplanar configurations 105
3.3.3 Signs of gradients three dimensional configurations . . 106
3.3.4 Degenerate configurations 107
3.4 Consequences for central configurations 109
3.4.1 Surprising regularity 109
3.4.2 Estimates on £CAL 112
3.4.3 Are there central configurations of these types? . 114
3.5 What can, and cannot, be 115
3.5.1 More central configurations 115
3.5.2 Masses and collinear central configurations 119
3.5.3 Masses and coplanar configurations 124
3.6 New kinds of constraints 126
3.7 Rings of Saturn . 130
3.7.1 Stability 131
3.7.2 More rings 133
3.7.3 Saturn, and some problems 136
CONTENTS ix
4 Collisions both real and imaginary 137
4.1 One body problem 138
4.1.1 Levi Civita's approach 140
4.1.2 Kustaanheimo and Stiefel's approach 141
4.1.3 Topological obstructions and hairy balls 143
4.1.4 Sundman's solution of the three body problem . 144
4.2 Sundman and the three body problem 147
4.2.1 Complex singularities? 147
4.2.2 Avoiding complex singularities 148
4.2.3 Singularities retaliate 149
4.3 Generalized Weierstrass Sundman theorem 150
4.3.1 A simple case the central force problem 151
4.3.2 Larger p values and "Black Holes" 151
4.3.3 Lagrange Jacobi equation 153
4.3.4 Proof of the Weierstrass Sundman Theorem 154
4.3.5 Bounded above means bounded below 159
4.3.6 Problems 161
4.3.7 An interesting historical footnote 161
4.4 Singularities an overview 162
4.4.1 Behavior of a singularity 163
4.4.2 Non collision singularities 165
4.4.3 Off to infinity 166
4.4.4 Problems 170
4.5 Rate of approach of collisions 172
4.5.1 General collisions 172
4.5.2 Tauberian Theorems 173
4.5.3 Proof of the theorem 179
4.6 Sharper asymptotic results 184
4.7 Spin, or no spin? 185
4.7.1 Using the angular momentum 187
4.7.2 The collinear case 189
4.8 Manifolds defined by collisions 191
4.8.1 Structure of collision sets 192
4.8.2 Proof of Theorem 4.18 193
4.9 Proof of the slowly varying assertion 195
4.9.1 Centers of mass 198
4.9.2 Back to the proof 201
4.9.3 The last steps 203
x CONTENTS
5 How likely is it? 207
5.1 Motivation 208
5.1.1 Idea of proof 209
5.1.2 Why do we need the Baire category statement? . 210
5.2 Proof: C is of first Baire category 210
5.2.1 Finding an appropriate C subset 211
5.2.2 A comment about the set of singularities 213
5.3 Proof: C is of Lebesgue measure zero 214
5.3.1 A common collision for k 3 particles 214
5.3.2 Lower dimensions, binary collisions, and other force
laws 217
5.4 Likelihood of non collision singularities 220
Bibliography 223
Index 232 |
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| genre_facet | Konferenzschrift 2002 Charleston Ill. |
| id | DE-604.BV020865133 |
| illustrated | Illustrated |
| index_date | 2024-07-02T13:24:13Z |
| indexdate | 2024-07-09T20:26:59Z |
| institution | BVB |
| isbn | 0821832506 |
| language | English |
| lccn | 2005041205 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014186978 |
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| physical | x, 235 p. Ill., graph. Darst. 26 cm |
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| publisher | American Mathematical Society |
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| series | CBMS regional conference series in mathematics |
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| spelling | Saari, Donald 1940- Verfasser (DE-588)124637418 aut Collisions, rings, and other Newtonian N-body problems Donald G. Saari Providence, R.I. American Mathematical Society 2005 x, 235 p. Ill., graph. Darst. 26 cm txt rdacontent n rdamedia nc rdacarrier CBMS regional conference series in mathematics 104 Includes bibliographical references (p. 223-231) and index Botsingen gtt Collisions (Astrophysique) - Congrès Newtonsystemen gtt Problème des N corps - Congrès Ringen (wiskunde) gtt Veel-deeltjes-systemen gtt Many-body problem Congresses Collisions (Astrophysics) Congresses Vielkörperproblem (DE-588)4078900-7 gnd rswk-swf Stoß (DE-588)4124258-0 gnd rswk-swf Astrophysik (DE-588)4003326-0 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2002 Charleston Ill. gnd-content Vielkörperproblem (DE-588)4078900-7 s Stoß (DE-588)4124258-0 s Astrophysik (DE-588)4003326-0 s b DE-604 CBMS regional conference series in mathematics 104 (DE-604)BV000004346 104 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014186978&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Saari, Donald 1940- Collisions, rings, and other Newtonian N-body problems CBMS regional conference series in mathematics Botsingen gtt Collisions (Astrophysique) - Congrès Newtonsystemen gtt Problème des N corps - Congrès Ringen (wiskunde) gtt Veel-deeltjes-systemen gtt Many-body problem Congresses Collisions (Astrophysics) Congresses Vielkörperproblem (DE-588)4078900-7 gnd Stoß (DE-588)4124258-0 gnd Astrophysik (DE-588)4003326-0 gnd |
| subject_GND | (DE-588)4078900-7 (DE-588)4124258-0 (DE-588)4003326-0 (DE-588)1071861417 |
| title | Collisions, rings, and other Newtonian N-body problems |
| title_auth | Collisions, rings, and other Newtonian N-body problems |
| title_exact_search | Collisions, rings, and other Newtonian N-body problems |
| title_exact_search_txtP | Collisions, rings, and other Newtonian N-body problems |
| title_full | Collisions, rings, and other Newtonian N-body problems Donald G. Saari |
| title_fullStr | Collisions, rings, and other Newtonian N-body problems Donald G. Saari |
| title_full_unstemmed | Collisions, rings, and other Newtonian N-body problems Donald G. Saari |
| title_short | Collisions, rings, and other Newtonian N-body problems |
| title_sort | collisions rings and other newtonian n body problems |
| topic | Botsingen gtt Collisions (Astrophysique) - Congrès Newtonsystemen gtt Problème des N corps - Congrès Ringen (wiskunde) gtt Veel-deeltjes-systemen gtt Many-body problem Congresses Collisions (Astrophysics) Congresses Vielkörperproblem (DE-588)4078900-7 gnd Stoß (DE-588)4124258-0 gnd Astrophysik (DE-588)4003326-0 gnd |
| topic_facet | Botsingen Collisions (Astrophysique) - Congrès Newtonsystemen Problème des N corps - Congrès Ringen (wiskunde) Veel-deeltjes-systemen Many-body problem Congresses Collisions (Astrophysics) Congresses Vielkörperproblem Stoß Astrophysik Konferenzschrift 2002 Charleston Ill. |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014186978&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004346 |
| work_keys_str_mv | AT saaridonald collisionsringsandothernewtoniannbodyproblems |