Notes on Dynamical Systems (Courant Lecture Notes)

This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. Jürgen Moser (1928–1999) was a professor at the Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jürgen Moser is the author of several books, among them Stable and Random Motions in Dynamical Systems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics. This Book Is An Introduction To The Field Of Dynamical Systems, In Particular, To The Special Class Of Hamiltonian Systems. The Authors Aimed At Keeping The Requirements Of Mathematical Techniques Minimal But Giving Detailed Proofs And Many Examples And Illustrations From Physics And Celestial Mechanics. After All, The Celestial N-body Problem Is The Origin Of Dynamical Systems And Gave Rise In The Past To Many Mathematical Developments. --publisher Description. Preface -- 1. Transformation Theory. Differential Equations And Vector Fields ; Variational Principles And Vector Fields ; Canonical Transformations ; Hamilton-jacobi Equations ; Integrals And Group Actions ; The So(4) Symmetry Of The Kepler Problem ; Symplectic Manifolds ; Hamiltonian Vector Fields On Symplectic Manifolds -- 2. Periodic Orbits. Poincaré's Perturbation Theory Of Periodic Orbits ; A Theorem By Lyapunov ; A Theorem By E. Hopf ; The Restricted 3-body Problem ; Reversible Systems ; The Plane 3- And 4-body Problems ; Poincaré-birkhoff Fixed Point Theorem ; Variations On The Fixed Point Theorems ; The Billiard Ball Problem ; A Theorem By Jacobowitz And Hartman ; Closed Geodesics On A Riemannian Manifold ; Periodic Orbits On A Convex Energy Surface ; Periodic Orbits Having Prescribed Periods -- 3. Integrable Hamiltonian Systems. A Theorem Of Arnold And Jost ; Delaunay Variables ; Integrals Via Asymptotics : The Störmer Problem ; The Toda Lattice ; Separation Of Variables ; Constrained Vector Fields ; Isospectral Deformations. Jürgen Moser, Eduard J. Zehnder. Includes Bibliographical References (p. 255-256). Preface Chapter 1. Transformation Theory 1.1. Differential Equations and Vector Fields 1.2. Variational Principles, Hamiltonian Systems 1.3. Canonical Transformations 1.4. Hamilton-Jacobi Equations 1.5. Integrals and Group Actions 1.6. The SO(4) Symmetry of the Kepler Problem 1.7. Symplectic Manifolds 1.8. Hamiltonian Vector Fields on Symplectic Manifolds Chapter 2. Periodic Orbits 2.1. Poincare's Perturbation Theory of Periodic Orbits 2.2. A Theorem by Lyapunov 2.3. A Theorem by E. Hopf 2.4. The Restricted 3-Body Problem 2.5. Reversible Systems 2.6. The Plane 3- and 4-Body Problems 2.7. Poincare-Birkhoff Fixed Point Theorem 2.8. Variations on the Fixed Point Theorems 2.9. The Billiard Ball Problem 2.10. A Theorem by Jacobowitz and Hartman 2.11. Closed Geodesies on a Riemannian Manifold 2.12. Periodic Orbits on a Convex Energy Surface 2.13. Periodic Orbits Having Prescribed Periods Chapter 3. Integrable Hamiltonian Systems 3.1. A Theorem of Arnold and Jost 3.2. Delaunay Variables 3.3. Integrals via Asymptotics; the Stormer Problem 3.4. The Toda Lattice 3.5. Separation of Variables 3.6. Constrained Vector Fields 3.7. Isospectral Deformations Bibliography This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. Jrgen Moser (1928-1999) was a professor at the Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jrgen Moser is the author of several books, among them Stable and Random Motions in Dynamical Systems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.