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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Heisenberg Algebras, Grassmannians and Isospectral Curves

Abstract
The connection between Heisenberg algebras and the geometry of branched covers of P1 by Riemann surfaces is discussed using infinite Grassmannians.

### Coadjoint Orbits, Spectral Curves and Darboux Coordinates

Abstract
For generic rational coadjoint orbits in the dual $$\tilde gl(r)^{ + *}$$ of the positive half of the loop algebra $$\tilde gl(r)^{ + *}$$, the natural divisor coordinates associated to the eigenvector line bundles over the spectral curves project to Darboux coordinates on the Gl(r)-reduced space. The geometry of the embedding of these curves in an ambient ruled surface suggests an intrinsic definition of symplectic structure on the space of pairs (spectral curves, duals of eigenvector line bundles) based on Serre duality. It is shown that this coincides with the reduced Kostant-Kirillov structure. For all Hamiltonians generating isospectral flows, these Darboux coordinates allow one to deduce a completely separated Liouville generating function, with the corresponding canonical transformation to linearizing variables identified as the Abel map.

### Hamiltonian Systems on the Jacobi Varieties

Abstract
New sequences of series of integrable Hamiltonian systems are obtained as the systems defined on the Jacobi varieties. Equations of Korteweg-de Vries (KdV) and inverse KdV type as well as sine-Gordon and nonlinear Schródinger equations together with the corresponding Hamiltonian systems are only initial members of the found sequences. We also investigate discrete systems associated with the integrable continuous systems.
Solomon J. Alber

### A Universal Reduction Procedure for Hamiltonian Group Actions

Abstract
We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the Marsden-Weinstein reduction is well-defined, the action is proper, and the preimage of a coadjoint orbit under the momentum mapping is closed, we show that universal reduction and Marsden-Weinstein reduction coincide. As an èxample, we explicitly construct the reduced spaces and their Poisson algebras for the spherical pendulum.
Judith M. Arms, Richard H. Cushman, Mark J. Gotay

### Linear Stability of a Periodic Orbit in the System of Falling Balls

Abstract
We study linear stability of a periodic orbit in the hamiltonian system with many degrees of freedom introduced in [W1].
Jian Cheng, Maciej P. Wojtkowski

### Birth and Death of Invariant Tori for Hamiltonian Systems

Abstract
The stability of invariant tori with (highly) irrational periods in the context of nearly integrable Hamiltonian systems and symplectic diffeomorphysms is considered. A theorem, based on computer-assisted implementations of recent KAM techniques, establishing the persistence of “golden-mean-tori” for “large” values of the nonlinearity parameter ε in two paradigmatic models is presented. Numerical investigations on the distribution of complex singularities in the parameter e indicates that the method of proof, which involves the explicit construction of approximating surfaces, is optimal at least in the models considered here.
Luigi Chierchia

### Dynamical Aspects of the Bidiagonal Singular Value Decomposition

Abstract
In this paper we describe some striking stability properties of the singular value decomposition (SVD) of a bidiagonal matrix and of the Hamiltonian flow which interpolates the standard SVD algorithm at integer times.
Percy Deift, James Demmel, Luen-Chau Li, Carlos Tomei

### The Rhomboidal Four Body Problem. Global Flow on the Total Collision Manifold

Abstract
In this work, we have considered a particular case of the planar four-body problem, obtained when the masses form a rhomboidal configuration. If we take the ratio of the masses α as a parameter, this problem is a one parameter family of non-integrable Hamiltonian systems with two degrees of freedom. We use the blow up method introduced by McGhee to study total collision. This singularity is replaced by an invariant two-dimensional manifold, called the total collision manifold. Using numerical methods we prove first that there are two equilibrium points for the flow on this manifold, and second, that there are only two values of a for which there is a connection between the invariant submanifolds of the equilibrium points. For these values of a the problem is not regularizable.

### Toward a Topological Classification of Integrable PDE’s

Abstract
We model Fomenko’s topological classification of 2 degree of freedom integrable stratifications in an infinite dimensional soliton system. Specifically, the analyticity of the Floquet discriminant Δ(q, ») in both of its arguments provides a transparent realization of a Bott function and of the remaining building blocks of the stratification; in this manner, Fomenko’s structure theorems are expressed through the inverse spectral transform. Thus, soliton equations are shown to provide natural representatives of the classification in the context of PDE’s.
Nicholas M. Ercolani, David W. McLaughlin

### Topological Classification of All Integrable Hamiltonian Differential Equations of General Type With Two Degrees of Freedom

Abstract
This paper is based on the series of lectures which were delivered by the author in 1989 at the Mathematical Sciences Research Institute in Berkeley. As part of the year-long 1988–1989 program in Symplectic Geometry and Mechanics, the Berkeley Mathematical Sciences Research Institute hosted a two-week workshop on the Geometry of Hamiltonian Systems on the period June 5 to June 16, 1989.
A. T. Fomenko

### Monotone Maps of × ∝ n and Their Periodic Orbits

Abstract
This article presents a theorem of existence of n + 1 (2 n if non degenerate) periodic orbits of any given rotation vector for a large class of exact symplectic maps of the n-dimensional annulus 핋 n × R n . This theorem is global and uses the discrete variational approach of Aubry.
Christophe Golé

### A Lower Bound for the Number of Fixed Points of Orientation Reversing Homeomorphisms

Abstract
Let h be an orientation reversing homeomorphism of the plane onto itself. Let X be a plane continuum, invariant under h. If X has at least 2 k invariant bounded complementary domains, then h has at least k + 2 fixed points in X.
Krystyna Kuperberg

### Invariant Tori and Cylinders for a Class of Perturbed Hamiltonian Systems

Abstract
We start with a relativistic model of the Kepler Problem, which is an isoenergetically non-degenerate central force problem in 2 dimensions. Then we prove the persistence of invariant cylinders and tori for a class of non Hamiltonian perturbations of this system.
Ernesto A. Lacomba, Jaume Llibre, Ana Nunes

### Modified Structures Associated with Poisson Lie Groups

Abstract
Given a Poisson Lie group, we show how to construct modified (Poisson) structures using suitable linear maps from the dual of the Lie algebra to the Lie algebra. An important class of such maps for which our construction works is given by morphisms of the corresponding Lie bialgebras. The twisted Poisson structures which arose in earlier work of the authors on Lax equations on a lattice are now identified as a special instance of this general construction.
Luen-Chau Li, Serge Parmentier

### Optimal Control of Deformable Bodies and Its Relation to Gauge Theory

Abstract
I investigate the question “What is the most efficient way for a deformable body to deform itself so as to achieve a desired reorientation?” I call this the Cat’s Problem, since itis the problem faced by the upside-down zero-angular momentum cat in freefall. In order of increasing generality, I show that the Cat’s Problem is a special case of problems which occur (1) in the geometry of principal bundles, (2)in subRiemannian geometry, and (3) in optimal control. Some model cases are explicitly solved in which the deformable body consists of a collection of point masses. In one of these models the principal bundle breaks down due to isotropy for the action of the rotation group. Nevertheless we are still able to obtain the general solution.
R. Montgomery

### The Augmented Divisor and Isospectral Pairs

Abstract
The Neumann system was shown by J. Moser (1979) to be isospectral with respect to a rank 2 perturbation of a diagonal matrix. In 1982 we showed that the Neumann system is isospectral with respect to a perturbation of a nilpotent 2 × 2 matrix. In this paper Baker functions are used to derive these matrices and to relate their eigenfunctions. I. M. Krichever used Baker functions to exhibit a homomorphism from a commutative ring of differential operators into a ring of functions. Moser’s matrix is derived in terms of a translation in the Jacobian Variety of the spectral curve and the 2 × 2 matrix is described using a rational extension of the Krichever homomorphism. The isospectral pair has three interesting geometric properties which are discussed below. They are related to the geometry of momentum mappings, to Moser’s work on the geometry of quadrics and to the Lie algebraic Euler equations.
Randolph James Schilling

### Symplectic Numerical Integration of Hamiltonian Systems

Abstract
In this paper I review several techniques to construct Symplectic Integration Algorithms. I also discuss algorithms for systems with other invariants such as Lie-Poisson structures reversible systems volume preserving flows. Numerical results are presented.
Clint Scovel

### Connections Between Critical Points in the Collision Manifold of the Planar 3-Body Problem

Abstract
We study the relative position of the invariant manifolds of the fixed points on the compactification of the so called non rotating triple collision manifold for the planar general three body problem. The results obtained allow to describe all the possible transition from the approach to triple collision to the escape from it. We also describe how these transitions change as a function of the masses of the three bodies in a domain of the space of masses and are completed by numerical simulations in the remainder set of masses.
C. Simó, A. Susín

### The Non-collision Sigularities of the 5 body Problem

Abstract
This paper is a survey on the author’s work on the non-collision singularities in the Newtonian n-body problem. The non-collision singularity in the n-body system corresponds to the solution which blow up to infinity in finite time. The question whether there exists such solution was first raised by Painleve in the last century and since then, it has been open. Here, we show that such solutions do exist in a 5-body problem. The method we use is based on careful analysis of near collisions orbits and McGehee’s technique of blowing up collision singularities.
Zhihong Xia
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