Dynamical Systems (Colloquium Publications)

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INTRODUCTION TO THE 1966 EDITION
PREFACE TO THE 1966 EDITION
PREFACE TO THE 1927 EDITION
CHAPTER 1: PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS
1. Introductory remarks
2. An existence theorem
3. A uniqueness theorem
4. Two continuity theorems
5. Some extensions
6. The principle of the conservation of energy. Conservation systems
7. Change of variables in conservative systems
8. Geometrical constraints
9. Internal characterization of Lagrangian systems
10. External characterization of Lagrangian systems
11. Dissipative systems
CHAPTER II: VARIATION AL PRINCIPLES AND APPLICATIONS
1. An algebraic variational principle
2. Hamilton\'s principle
3. The principle of least action
4. Normal form (two degrees of freedom)
5. Ignorable coodinates
6. The method of multipliers
7. The general integral linear in the velocities
8. Conditional integrals linear in the velocities
9. Integrals quadratic in the velocities
10. The Hamiltonian equations
II. Transfonnation of the Hamiltonian equations
12. The Pfaffian equations
13. On the significance of variational principles
CHAPTER III: FORMAL ASPECTS OF DYNAMICS
1. Introductory remarks
2. The formal group
3. Formal solutions
4. The equilibrium problem
5. The generalized equilibrium problem
6. On the Hamiltonian multipliera
7. Normalization of H₂
8. The Hamiltonian equilibrium problem
9. Generalization of the Hamiltonian problem
10. On the pfaffian multipliers
11. Preliminary normalization in pfaffian problem
12. The Pfaffian equilibrium problem
13. Generalization of the Pfaffian problem
CHAPTER IV: STABILITY OF PERIODIC MOTIONS
1. On the reduction to generalized equilibrium
2. Stability of Pfaffian systems
3. Instability of pfaffian systems
4. Complete stability
5. Normal form for completely stable systems
6. Proof of the lemma of section 5
7. Reversibility and complete stability
8. Other types of stability
CHAPTER V: EXISTENCE OF PERIODIC MOTIONS
1. Role of the periodic motions
2. An example
3. The minimum method
4. Application to symmetric case
5. Whittaker\'s criterion and analogous results
6. The minimax method
7. Application to exceptional case
8. The extensions by Morse
9. The method of analytic continuation
10. The transfonnation method of Poincaré
11. An example
CHAPTER VI: APPLICATION OF POINCARE\'S GEOMETRIC THEOREM
1. Periodic motions near generalized equilibrium (m = 1)
2. Proof of the lemma of section 1
3. Periodic motions near a periodic motion (m = 2)
4. Some remarks
5. The geometric theorem of Poincaré
6. The billiard ball problem
7. The corresponding transformation T
8. Area-preserving property of T
9. Applications to billiard ball problem
10. The geodesic problem. Construction of a transformation TT*
11. Application of Poincaré\'s theorem to geodesic problem
CHAPTER VII: GENERAL THEORY OF DYNAMICAL SYSTEMS
1. Introductory remarks
2. Wandering and non-wandering motions
3. The sequence M, M₁, M₂
4. Some properties of the central motions
5. Concerning the role of the central motions
6. Groups of motions
7. Recurrent motions
8. Arbitrary motions and the recurrent motions
9. Density of the special central motions
10. Recurrent motions and semi-asymptotic central motions
11. Transitivity and intransitivity
CHAPTER VIII: THE CASE OF TWO DEGREES OF FREEDOM
1. Formal classification of invariant points
2. Distribution of periodic motions of stable type
3. Distribution of quasi-periodic motions
4. Stability and instability
5. The stable case. Zones of instability
6. A criterion for stability
7. The problem of stability
8. The unstable case. Asymptotic families
9. Distribution of motions asymptotic to periodic motions
10. On other types of motion
11. A transitive dynamical problem
12. An integrable case
13. The concept of integrability
CHAPTER IX: THE PROBLEM OF THREE BODIES
1. Introductory remarks
2. The equations of motion and the classical integrals
3. Reduction to the 12th order
4. Lagrange\'s equality
5. Sundman\'s inequality
6. The possibility of collision
7. Indefinite continuation of the motions
8. Further properties of the motions
9. On a result of Sundman
10. The reduced manifold M₇ of states of motion
11. Types of motion in M₇
12. Extension to n > 3 bodies and more general laws of force
ADDENDUM
FOOTNOTES
BIBLIOGRAPHY
INDEX