Dynamical systems

Автор(ы):Birkhoff, George D.
06.10.2007
Год изд.:1966
Издание:2
Описание: Many mathematicians will welcome the new edition of G. D. Birkhoffs book on Dynamical Systems. It represents essentially a continuation of Poincare's profound and extensive work on Celestial Mechanics. Altogether Birkhoff was strongly influenced by Poincare and devoted a major part of his mathematical work to subjects arising from Poincare's tradition. The present book contains Birkhoff s views and ideas of his earlier period of life—it appeared when Birkhoff was 43. To the modern reader the style of this book may appear less formal and rigorous than it is now customary. But just the informal and lively manner of writing has been inspiring to many mathematicians. The effect of this inspiration is visible in a number of later papers. For example, Morse's theory on geodesies on a closed manifold originated directly in Birkhoffs ideas in dynamical systems. The recent work by Anosov on U-Systems answers the question of ergodicity and density of periodic solutions for a wide class of differential equations—a problem which in Birkhoff s book was studied for a single model system. These and other examples (given below) justify the hope that the reprinting of this book again will stimulate further progress.
Оглавление:
Dynamical systems — обложка книги.
CHAPTER I
PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS
  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. Conservation 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]
  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
  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 (?) [78]
  8. The Hamiltonian equilibrium problem [82]
  9. Generalization of the H amiltonian 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]
  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 Poincare [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 Т [171]
  8. Area-preserving property of Т [173]
  9. Applications to billiard ball problem [176]
  10. The geodesic problem. Construction of a transformation 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, (?),(?) [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]
CHAPTER VIII
THE CASE OF TWO DEGREES OF FREEDOM
  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 (?) of states of motion [283]
  11. Types of motion in (?) [288]
  12. Extension to n > 3 bodies and more general laws offeree [291]
ADDENDUM [293]
FOOTNOTES [296]
BIBLIOGRAPHY [300]
INDEX [303]
Формат: djvu
Размер:7072827 байт
Язык:ENG
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