Geometric Mechanics and Symmetry: The Peyresq Lectures (London Mathematical Society Lecture Note Series, Vol. 306)

The lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems. They succinctly cover an unparalleled range of topics from the basic concepts of symplectic and Poisson geometry, through integrable systems, KAM theory, fluid dynamics, and symmetric bifurcation theory. The lectures are based on summer schools for graduate students and postdocs and provide complementary and contrasting viewpoints of key topics: the authors cut through an overwhelming amount of literature to show young mathematicians how to get to the core of the various subjects and thereby enable them to embark on research careers. Cover; Title; Copyright; Contents; List of contributors; Preface; I Stability in Hamiltonian Systems: Applications to the restricted three-body problem (K.R. Meyer); 1. Introduction; 2. Restricted three-body problem; 3. Relative equilibria; 4. Linear Hamiltonian Systems; 5. Liapunov's and Chetaev's theorems; 6. Applications to the restricted problem; 7. Normal forms; 8. Poincare sections; 9. The twist map and Arnold's stability theorem; References; II A Crash Course in Geometric Mechanics (T.S. Ratiu); 2. Hamiltonian Formalism; 3. Lagrangian Formalism; 4. Poisson Manifolds; 5. Momentum Maps 1.2. U(2) momentum map1.3. Hopf fibration; 1.4. Normalization; 1.5. Normalization of the Henon-Heiles hamiltonian; A. Comments on Lecture I. The Henon-Heiles system; A.1. Invariants and integrity basis; A.2. Qualitative analysis of the reduced system; A.3. Normal form and remarks on further analysis; 2. Lectures III and V. The Euler top; 2.1. Preliminaries on the rotation group; 2.2. Traditional derivation of the equations of motion; 2.3. Qualitative behavior of solutions of Euler's equations; 2.4. Quantitative behavior of solutions of Euler's equations; 2.5. The Euler-Arnol'd equations V Survey on dissipative KAM theory including quasi-periodic bifurcation theory (H. Broer)1. Introduction; 2. Quasi-periodic attractors; 3. Towards a KAM theory of vector fields; 4. The normal linear part of quasi-periodic tori; 5. Elements of quasi-periodic bifurcation theory; 6. Concluding remarks; References; Appendix; VI Symmetric Hamiltonian Bifurcations (J.A. Montaldi); 1. Introduction; PART I: LOCAL DYNAMICS NEAR EQUILIBRIA; 2. Nonlinear normal modes; 3. Generic bifurcations near equilibria; 4. Hamiltonian-Hopf bifurcation; PART II: LOCAL DYNAMICS NEAR RELATIVE EQUILIBRIA 6. Lie-Poisson and Euler-Poincare Reduction7. Symplectic Reduction; References; III The Euler-Poincare variational framework for modeling fluid dynamics (D.D. Holm); 1. The problem of ocean circulation & global climate; 2. Euler-Poincare fluid dynamics; 3. Applications of the Euler-Poincare theorem in GFD; 4. Lagrangian reduction and EP turbulence closures; 5. Pulsons and peakons; 6. Momentum filaments and surfaces; References; IV No polar coordinates (R.H. Cushman); Avant Propos; 1. Lectures I and II. The two dimensional harmonic oscillator; 1.1. The harmonic oscillator 2.6. Abstract derivation of equations of motionB. Comments on lecture III; B.1. The herpolhode; B.2. Finite symmetries; 3. Lecture IV. The spherical pendulum and monodromy; 3.1. The unconstrained system; 3.2. Constrained system; 3.3. Reduction of S1 symmetry; 3.4. Analysis of the reduced system; 3.5. Reconstruction; 3.6. Monodromy; C. Comments on Lecture IV; C.2. Geometric analysis on P_/Z2; D. Homework problem. A degenerately pinched 2-torus; D.1. Introduction; D.2. A model system; D.3. A special case; D.4. An example of a degenerate heteroclinic cycle; References

Geometric mechanics borders pure and applied mathematics and incorporates such disciplines as differential geometry, Hamiltonian mechanics and integrable systems. The source of this collection is the summer school on Geometric Mechanics and Symmetry organized by James Montaldi and Tudor Ratiu. Written with significant input from the participants at the conference, these lecture notes are geared towards fulfilling the needs of graduate students through their attention to detail.

These lectures are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems. They cover a range of topics from the basic concepts of symplectic and Poisson geometry, through integrable systems, KAM theory, fluid dynamics, and symetric bifurcation theory As participants in the MASIE-project, we attended the summer school Mechanics and Symmetry in Peyresq, France, during the first two weeks of September 2000.