Dynamical systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
Providence, RI
American Math. Soc.
1927 [erschienen] 1966
|
| Edition: | 1. print. of rev. ed. |
| Series: | American Mathematical Society colloquium publications
9 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 305 S. |
Staff View
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| 245 | 1 | 0 | |a Dynamical systems |c by George D. Birkhoff |
| 250 | |a 1. print. of rev. ed. | ||
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1927 [erschienen] 1966 | |
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Record in the Search Index
| _version_ | 1804136000357662720 |
|---|---|
| adam_text | TABLE OF CONTENTS
Introduction to the 1966 Edition iii
Preface to the 1966 Edition iv
Preface to the 1927 Edition vi
CHAPTER I
PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS
Page
1. Introductory remarks 1
2. An existence theorem 1
3. A uniqueness theorem 5
4. Two continuity theorems 6
5. Some extensions 10
6. The principle of the conservation of energy. Con¬
servation systems 14
7. Change of variables in conservative systems 19
8. Geometrical constraints 22
9. Internal characterization of Lagrangian systems 23
10. External characterization of Lagrangian systems 25
11. Dissipative systems 31
CHAPTER II
VARIATIONAL PRINCIPLES AND APPLICATIONS
1. An algebraic variational principle 33
2. Hamilton s principle 34
3. The principle of least action 36
4. Normal form (two degrees of freedom) 39
5. Ignorable coodinates 40
6. The method of multipliers 41
7. The general integral linear in the velocities 44
8. Conditional integrals linear in the velocities 45
ix
X CONTENTS
9. Integrals quadratic in the velocities 48
10. The Hamiltonian equations 50
11. Transformation of the Hamiltonian equations 53
12. The Pfaffian equations 55
13. On the significance of variational principles 55
CHAPTER III
FORMAL ASPECTS OF DYNAMICS
Page
1. Introductory remarks 59
2. The formal group 60
3. Formal solutions 63
4. The equilibrium problem 67
5. The generalized equilibrium problem 71
6. On the Hamiltonian multipliers 74
7. Normalization of if2 78
8. The Hamiltonian equilibrium problem 82
9. Generalization of the Hamiltonian problem 85
10. On the Pfaffian multipliers 89
11. Preliminary normalization in Pfaffian problem 91
12. The Pfaffian equilibrium problem 93
13. Generalization of the Pfaffian problem 94
CHAPTER IV
STABILITY OF PERIODIC MOTIONS
1. On the reduction to generalized equilibrium 97
2. Stability of Pfaffian systems 100
3. Instability of Pfaffian systems 105
4. Complete stability 105
5. Normal form for completely stable systems 109
6. Proof of the lemma of section 5 114
7. Reversibility and complete stability 115
8. Other types of stability 121
CHAPTER V
EXISTENCE OF PERIODIC MOTIONS
1. Role of the periodic motions 123
2. An example 124
CONTENTS xi
3. The minimum method 128
4. Application to symmetric case 130
5. Whittaker s criterion and analogous results 132
6. The minimax method 133
7. Application to exceptional case 135
8. The extensions by Morse 139
9. The method of analytic continuation 139
10. The transformation method of Poincar6 143
11. An example 146
CHAPTER VI
APPLICATION OF POINCARE S GEOMETRIC THEOREM
1. Periodic motions near generalized equilibrium (m = 1) 150
2. Proof of the lemma of section 1 154
3. Periodic motions near a periodic motion (m = 2) 159
4. Some remarks 162
5. The geometric theorem of Poincare 165
6. The billiard ball problem 169
7. The corresponding transformation T. 171
8. Area preserving property of T 173
9. Applications to billiard ball problem 176
10. The geodesic problem. Construction of a transform¬
ation TT* 180
11. Application of Poincare s theorem to geodesic
problem 185
CHAPTER VII
GENERAL THEORY OF DYNAMICAL SYSTEMS
1. Introductory remarks 189
2. Wandering and non wandering motions 190
3. The sequence M, Mj, M2 193
4. Some properties of the central motions 195
5. Concerning the role of the central motions 197
6. Groups of motions 197
7. Recurrent motions 198
8. Arbitrary motions and the recurrent motions 200
9. Density of the special central motions 202
10. Recurrent motions and semi asymptotic central
motions 204
11. Transitivity and intransitivity 205
Xli CONTENTS
CHAPTER VIII
THE CASE OF TWO DEGREES OF FREEDOM
Page
1. Formal classification of invariant points 209
2. Distribution of periodic motions of stable type 215
3. Distribution of quasi periodic motions 218
4. Stability and instability 220
5. The stable case. Zones of instability 221
6. A criterion for stability 226
7. The problem of stability 227
8. The unstable case. Asymptotic families 227
9. Distribution of motions asymptotic to periodic
motions 231
10. On other types of motion 237
11. A transitive dynamical problem 238
12. An integrable case 248
13. The concept of integrability 255
CHAPTER IX
THE PROBLEM OF THREE BODIES
1. Introductory remarks 260
2. The equations of motion and the classical integrals 261
3. Reduction to the 12th order 263
4. Lagrange s equality 264
5. Sundman s inequality 265
6. The possibility of collision 267
7. Indefinite continuation of the motions 270
8. Further properties of the motions 275
9. OnaresultofSundman 283
10. The reduced manifold M7 of states of motion 283
11. Types of motion in Af 7 288
12. Extension to n 3 bodies and more general laws
offeree 291
Addendum 293
Footnotes 296
Bibliography 300
Index 303
|
| adam_txt |
TABLE OF CONTENTS
Introduction to the 1966 Edition iii
Preface to the 1966 Edition iv
Preface to the 1927 Edition vi
CHAPTER I
PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS
Page
1. Introductory remarks 1
2. An existence theorem 1
3. A uniqueness theorem 5
4. Two continuity theorems 6
5. Some extensions 10
6. The principle of the conservation of energy. Con¬
servation systems 14
7. Change of variables in conservative systems 19
8. Geometrical constraints 22
9. Internal characterization of Lagrangian systems 23
10. External characterization of Lagrangian systems 25
11. Dissipative systems 31
CHAPTER II
VARIATIONAL PRINCIPLES AND APPLICATIONS
1. An algebraic variational principle 33
2. Hamilton's principle 34
3. The principle of least action 36
4. Normal form (two degrees of freedom) 39
5. Ignorable coodinates 40
6. The method of multipliers 41
7. The general integral linear in the velocities 44
8. Conditional integrals linear in the velocities 45
ix
X CONTENTS
9. Integrals quadratic in the velocities 48
10. The Hamiltonian equations 50
11. Transformation of the Hamiltonian equations 53
12. The Pfaffian equations 55
13. On the significance of variational principles 55
CHAPTER III
FORMAL ASPECTS OF DYNAMICS
Page
1. Introductory remarks 59
2. The formal group 60
3. Formal solutions 63
4. The equilibrium problem 67
5. The generalized equilibrium problem 71
6. On the Hamiltonian multipliers 74
7. Normalization of if2 78
8. The Hamiltonian equilibrium problem 82
9. Generalization of the Hamiltonian problem 85
10. On the Pfaffian multipliers 89
11. Preliminary normalization in Pfaffian problem 91
12. The Pfaffian equilibrium problem 93
13. Generalization of the Pfaffian problem 94
CHAPTER IV
STABILITY OF PERIODIC MOTIONS
1. On the reduction to generalized equilibrium 97
2. Stability of Pfaffian systems 100
3. Instability of Pfaffian systems 105
4. Complete stability 105
5. Normal form for completely stable systems 109
6. Proof of the lemma of section 5 114
7. Reversibility and complete stability 115
8. Other types of stability 121
CHAPTER V
EXISTENCE OF PERIODIC MOTIONS
1. Role of the periodic motions 123
2. An example 124
CONTENTS xi
3. The minimum method 128
4. Application to symmetric case 130
5. Whittaker's criterion and analogous results 132
6. The minimax method 133
7. Application to exceptional case 135
8. The extensions by Morse 139
9. The method of analytic continuation 139
10. The transformation method of Poincar6 143
11. An example 146
CHAPTER VI
APPLICATION OF POINCARE'S GEOMETRIC THEOREM
1. Periodic motions near generalized equilibrium (m = 1) 150
2. Proof of the lemma of section 1 154
3. Periodic motions near a periodic motion (m = 2) 159
4. Some remarks 162
5. The geometric theorem of Poincare 165
6. The billiard ball problem 169
7. The corresponding transformation T. 171
8. Area preserving property of T 173
9. Applications to billiard ball problem 176
10. The geodesic problem. Construction of a transform¬
ation TT* 180
11. Application of Poincare's theorem to geodesic
problem 185
CHAPTER VII
GENERAL THEORY OF DYNAMICAL SYSTEMS
1. Introductory remarks 189
2. Wandering and non wandering motions 190
3. The sequence M, Mj, M2 193
4. Some properties of the central motions 195
5. Concerning the role of the central motions 197
6. Groups of motions 197
7. Recurrent motions 198
8. Arbitrary motions and the recurrent motions 200
9. Density of the special central motions 202
10. Recurrent motions and semi asymptotic central
motions 204
11. Transitivity and intransitivity 205
Xli CONTENTS
CHAPTER VIII
THE CASE OF TWO DEGREES OF FREEDOM
Page
1. Formal classification of invariant points 209
2. Distribution of periodic motions of stable type 215
3. Distribution of quasi periodic motions 218
4. Stability and instability 220
5. The stable case. Zones of instability 221
6. A criterion for stability 226
7. The problem of stability 227
8. The unstable case. Asymptotic families 227
9. Distribution of motions asymptotic to periodic
motions 231
10. On other types of motion 237
11. A transitive dynamical problem 238
12. An integrable case 248
13. The concept of integrability 255
CHAPTER IX
THE PROBLEM OF THREE BODIES
1. Introductory remarks 260
2. The equations of motion and the classical integrals 261
3. Reduction to the 12th order 263
4. Lagrange's equality 264
5. Sundman's inequality 265
6. The possibility of collision 267
7. Indefinite continuation of the motions 270
8. Further properties of the motions 275
9. OnaresultofSundman 283
10. The reduced manifold M7 of states of motion 283
11. Types of motion in Af 7 288
12. Extension to n 3 bodies and more general laws
offeree 291
Addendum 293
Footnotes 296
Bibliography 300
Index 303 |
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| index_date | 2024-07-02T16:12:27Z |
| indexdate | 2024-07-09T20:49:26Z |
| institution | BVB |
| language | Undetermined |
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| physical | 305 S. |
| publishDate | 1927 |
| publishDateSearch | 1966 |
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| publisher | American Math. Soc. |
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| series | American Mathematical Society colloquium publications |
| series2 | American Mathematical Society colloquium publications |
| spelling | Birkhoff, George David 1884-1944 Verfasser (DE-588)118663399 aut Dynamical systems by George D. Birkhoff 1. print. of rev. ed. Providence, RI American Math. Soc. 1927 [erschienen] 1966 305 S. txt rdacontent n rdamedia nc rdacarrier American Mathematical Society colloquium publications 9 Mechanik (DE-588)4038168-7 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s DE-604 Mechanik (DE-588)4038168-7 s 1\p DE-604 American Mathematical Society colloquium publications 9 (DE-604)BV035417609 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015241046&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Birkhoff, George David 1884-1944 Dynamical systems American Mathematical Society colloquium publications Mechanik (DE-588)4038168-7 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4038168-7 (DE-588)4013396-5 |
| title | Dynamical systems |
| title_auth | Dynamical systems |
| title_exact_search | Dynamical systems |
| title_exact_search_txtP | Dynamical systems |
| title_full | Dynamical systems by George D. Birkhoff |
| title_fullStr | Dynamical systems by George D. Birkhoff |
| title_full_unstemmed | Dynamical systems by George D. Birkhoff |
| title_short | Dynamical systems |
| title_sort | dynamical systems |
| topic | Mechanik (DE-588)4038168-7 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Mechanik Dynamisches System |
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