His research in dynamics constitutes the middle period of Birkhoff's scientific career, that of maturity and greatest power. --Yearbook of the American Philosophical Society The author's great book ... is well known to all, and the diverse active modern developments in mathematics which have been inspired by this volume bear the most eloquent testimony to its quality and influence. --Zentralblatt MATH In 1927, G. D. Birkhoff wrote a remarkable treatise on the theory of dynamical systems that would inspire many later mathematicians to do great work. To a large extent, Birkhoff was writing about his own work on the subject, which was itself strongly influenced by Poincare's approach to dynamical systems. With this book, Birkhoff also demonstrated that the subject was a beautiful theory, much more than a compendium of individual results. The influence of this work can be found in many fields, including differential equations, mathematical physics, and even what is now known as Morse theory. The present volume is the revised 1966 reprinting of the book, including a new addendum, some footnotes, references added by Jurgen Moser, and a special preface by Marston Morse. Although dynamical systems has thrived in the decades since Birkhoff's book was published, this treatise continues to offer insight and inspiration for still more generations of mathematicians.
Author(s): George D. Birkhoff
Publisher: American Mathematical Society
Year: 1927
Language: English
Pages: 318
Cover......Page 1
Title......Page 2
LCCN 28-28411 ......Page 3
INTRODUCTION TO THE 1966 EDITION ......Page 4
A PREFACE TO THE 1966 EDITION ......Page 5
PREFACE TO THE 1927 EDITION ......Page 7
TABLE OF CONTENTS ......Page 9
2. An existence theorem. ......Page 13
3. A uniqueness theorem. ......Page 17
4. Two continuity theorems. ......Page 18
5. Some extensions. ......Page 22
6. The principle of the conservation of energy.* ......Page 26
7. Change of variables in conservative systems. ......Page 31
8. Geometrical constraints. ......Page 34
9. Internal characterization of Lagrangtan systems. ......Page 35
10. External characterization of Lagrangian systems......Page 37
11. Dissipative systems. ......Page 43
1. An algebraic variational principle. ......Page 45
2. Hamilton's principle. ......Page 46
3. The principle of least action. ......Page 48
4. Normal form (two degrees of freedom). ......Page 51
5. Ignorable coordinates. ......Page 52
6. The method of multipliers. ......Page 53
7. The general integral linear in the velocities. ......Page 56
8. Conditional integrals linear ......Page 57
9. Integrals quadratic in the velocities. ......Page 60
10. The Hamiltonian equations. ......Page 62
11. Transformation of the Hamiltonian equations. ......Page 65
13. On the significance of variational principles. ......Page 67
1. Introductory remarks. ......Page 71
2. The formal group. ......Page 72
3. Formal solutions. ......Page 75
4. The equilibrium problem. ......Page 79
5. The generalized equilibrium problem. ......Page 83
6. On the Hamiltonian multipliers. ......Page 86
7. Normalization of H*. ......Page 90
8. The Hamiltonian equilibrium problem. ......Page 94
9. Generalization of the Hsmiltonian problem......Page 97
10. On the Pfaffian multipliers. ......Page 101
ii. Preliminary normalization in Pfaffian problem. ......Page 103
12. The Pfaffian equilibrium problem. ......Page 105
13. Generalization of the Pfaffian problem. ......Page 106
1. On the reduction to generalized equilibrium. ......Page 109
2. Stability of Pfaffian systems. ......Page 112
4. Complete stability* ......Page 117
5. Normal form for completely stable systems. ......Page 121
6. Proof of the lemma of section 5. ......Page 126
7. Reversibility and complete stability. ......Page 127
8. Other types of stability. ......Page 133
1. Role of the periodic motions. ......Page 135
2. An example. ......Page 136
3. The minimum method. ......Page 140
4. Application to symmetric case. ......Page 142
5. Whittaker's criterion and analogous results. ......Page 144
6. The minimax method. ......Page 145
7. Application to exceptional case. ......Page 147
9. The method of analytic continuation. ......Page 151
10. The transformation method of Poincare. ......Page 155
11. An example. ......Page 158
i. Periodic motions near generalized equilibrium (m=1)......Page 162
2. Proof of the lemma of section 1 ......Page 166
3. Periodic motions near a periodic motion (m = 2). ......Page 171
4. Some remarks. ......Page 174
5. The geometric theorem of Poincare.* ......Page 177
6. The billiard ball problem.! ......Page 181
7. The corresponding transformation T. ......Page 183
8. Area-preserving property of T. ......Page 185
9. Applications to billiard ball problem. ......Page 188
10. The geodesic problem. Construction of a transformation TT*......Page 192
11. Application of Poincare's theorem to problem. ......Page 197
1. Introductory remarks......Page 201
2. Wandering and non-wandering motions. ......Page 202
3. The sequence M, M1, M2, ... ......Page 205
4. Some properties of the central motions. ......Page 207
6. Groups of motions ......Page 209
7. Recurrent motions. ......Page 210
8. Arbitrary motions and the recurrent ......Page 212
9. Density of the special central motions. ......Page 214
10. Recurrent motions and semi-asymptotic central motions......Page 216
11. Transitivity and intransitivity. ......Page 217
1. Formal classification of periodic motions. ......Page 221
2. Distribution of periodic motions of stable type. ......Page 227
3. Distribution of quasi-periodic motions. ......Page 230
4. Stability and instability. ......Page 232
5. The stable case. Zones of instability. ......Page 233
6. A criterion for stability. ......Page 238
8. The unstable case. ......Page 239
9. Distribution of motions asymptotic to periodic motions......Page 243
10. On other types of motion. ......Page 249
11. A transitive dynamical problem. ......Page 250
12. An integrable case. ......Page 260
13. The concept of integrability. ......Page 267
1. Introductory remarks. ......Page 272
2. The equations of motion and the classical integrals......Page 273
3. Reduction to the 12th order. ......Page 275
4. Lagrange's equality. ......Page 276
5. Sundman's inequality. ......Page 277
6. The possibility of collision. ......Page 279
7. Indefinite continuation of the motions. ......Page 282
8. Further properties of the motions. ......Page 287
10. The reduced manifold M-, of states of motion. ......Page 295
11. Types of motion in M......Page 300
12. Extension to n>3 bodies and more general laws of force......Page 303
ADDENDUM ......Page 305
FOOTNOTES ......Page 308
BIBLIOGRAPHY ......Page 312
INDEX......Page 315
Back Cover......Page 318
