The Geometry Of Hamiltonian Systems: Proceedings Of A Workshop Held June 5-16, 1989 (mathematical Sciences Research Institute Publications)

The papers in this volume are an outgrowth of the lectures and informal discussions that took place during the workshop on "The Geometry of Hamiltonian Systems" which was held at MSRl from June 5 to 16, 1989. It was, in some sense, the last major event of the year-long program on Symplectic Geometry and Mechanics. The emphasis of all the talks was on Hamiltonian dynamics and its relationship to several aspects of symplectic geometry and topology, mechanics, and dynamical systems in general. The organizers of the conference were R. Devaney (co-chairman), H. Flaschka (co-chairman), K. Meyer, and T. Ratiu. The entire meeting was built around two mini-courses of five lectures each and a series of two expository lectures. The first of the mini-courses was given by A. T. Fomenko, who presented the work of his group at Moscow University on the classification of integrable systems. The second mini­ course was given by J. Marsden of UC Berkeley, who spoke about several applications of symplectic and Poisson reduction to problems in stability, normal forms, and symmetric Hamiltonian bifurcation theory. Finally, the two expository talks were given by A. Fathi of the University of Florida who concentrated on the links between symplectic geometry, dynamical systems, and Teichmiiller theory. Front Matter....Pages i-x Heisenberg Algebras, Grassmannians and Isospectral Curves....Pages 1-7 Coadjoint Orbits, Spectral Curves and Darboux Coordinates....Pages 9-21 Hamiltonian Systems on the Jacobi Varieties....Pages 23-32 A Universal Reduction Procedure for Hamiltonian Group Actions....Pages 33-51 Linear Stability of a Periodic Orbit in the System of Falling Balls....Pages 53-71 Birth and Death of Invariant Tori for Hamiltonian Systems....Pages 73-80 Dynamical Aspects of the Bidiagonal Singular Value Decomposition....Pages 81-96 The Rhomboidal Four Body Problem. Global Flow on the Total Collision Manifold....Pages 97-110 Toward a Topological Classification of Integrable PDE’s....Pages 111-129 Topological Classification of All Integrable Hamiltonian Differential Equations of General Type With Two Degrees of Freedom....Pages 131-339 Monotone Maps of $${\mathbb{T}^n}$$ × ∝ n and Their Periodic Orbits....Pages 341-366 A Lower Bound for the Number of Fixed Points of Orientation Reversing Homeomorphisms....Pages 367-371 Invariant Tori and Cylinders for a Class of Perturbed Hamiltonian Systems....Pages 373-385 Modified Structures Associated with Poisson Lie Groups....Pages 387-401 Optimal Control of Deformable Bodies and Its Relation to Gauge Theory....Pages 403-438 The Augmented Divisor and Isospectral Pairs....Pages 439-461 Symplectic Numerical Integration of Hamiltonian Systems....Pages 463-496 Connections Between Critical Points in the Collision Manifold of the Planar 3-Body Problem....Pages 497-518 The Non-collision Sigularities of the 5 body Problem....Pages 519-527 The papers in this volume are an outgrowth of the lectures and informal discussions that took place during the workshop on "The Geometry of Hamiltonian Systems" which was held at MSRl from June 5 to 16, 1989. It was, in some sense, the last major event of the year-long program on Symplectic Geometry and Mechanics. The emphasis of all the talks was on Hamiltonian dynamics and its relationship to several aspects of symplectic geometry and topology, mechanics, and dynamical systems in general. The organizers of the conference were R. Devaney (co-chairman), H. Flaschka (co-chairman), K. Meyer, and T. Ratiu. The entire meeting was built around two mini-courses of five lectures each and a series of two expository lectures. The first of the mini-courses was given by A.T. Fomenko, who presented the work of his group at Moscow University on the classification of integrable systems. The second miniƯ course was given by J. Marsden of UC Berkeley, who spoke about several applications of symplectic and Poisson reduction to problems in stability, normal forms, and symmetric Hamiltonian bifurcation theory. Finally, the two expository talks were given by A. Fathi of the University of Florida who concentrated on the links between symplectic geometry, dynamical systems, and Teichmiiller theory The papers in this volume are an outgrowth of some of the lectures and informal discussions that took place during the workshop on the geometry of Hamiltonian systems, held at the MSRI in Berkeley in June of 1989. The emphasis of all the talks was on Hamiltonian dynamics and its relationship to several aspects of symplectic geometry and topology, mechanics, numerical simulations and dynamical systems in general. The articles are of differing lengths and scopes; some are research announcements while others are surveys of particularly active areas of interest where the results can only be found in scattered research articles and preprints. In- cluded in the book is A.T. Fomenko's survey of the classification of integrable systems.