Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases.

This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many

Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinear-waves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the three-wave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.

This paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivariant Hopf bifurcation on C 4 . The normal form has a symmetry group given by the semi-direct product of D2 with T 2 \Theta S 1 . The Hamiltonian part of the normal form is completely integrable and may be analyzed using a system of invariants. The idea of the paper is to perturb relative equilibria in this singular Hamiltonian limit to obtain new three frequency solutions to the full normal form for parameter values near the Hamiltonian limit. The solutions obtained have fully broken symmetry, that is, they do not lie in fixed point subspaces. The methods developed in this paper allow one to determine the stability of this new branch of solutions. An example shows that the branch of three-tori can be stable. 1 Introduction A standard approach in the bifurcation analysis of spatially-extended systems (such as RayleighBenard convection in an infinite plane) is to res...

We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ɛ, λ), where ɛ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by three-dimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or off-center displacement. 1

This paper delineates a class of time-periodically perturved evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form ˙x = f0(x) + εf1(x, t), where ˙x = f0(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam. 1

The integrable structure of the three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian are associated with the three-wave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigid-body or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasi-phase-matching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.

Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. We use a special set of action-angle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced flow of the normal form yields information on the original system. We illustrate the results by analysing the family of (arbitrary) axially symmetric cubic potentials. 1 Introduction One of the few methods that are available to study Hamiltonian systems is to find an integrable system that is close to it and to consider the former as a perturbation of the latter. In case the integrable system is non-degenerate, the flow of this system makes the phase space a ramified torus bundle. For instance, in three degrees of ...

This paper was also one of the first to notice deep links between reduction and integrable systems, a subject continued by, for example, Bobenko, Reyman and Semenov-Tian-Shansky [1989].

In this Preface, we make some brief remarks about the history, theory and applications of symplectic reduction. We concentrate on developments surrounding our paper Marsden and Weinstein [1974] and the closely related work of Meyer [1973], so the reader may find some important references omitted. This is inevitable in a subject that has grown so large and has penetrated so deeply both pure and applied mathematics, as well as into engineering and theoretical physics. We thank Klaas Landsman for the invitation to write these introductory remarks for this exciting book. We hope that they will be especially useful for younger workers in the area. Some of this preface is taken, with some revision, from an introductory section in Marsden, Ratiu and Scheurle [2000]. We would like to thank Tudor Ratiu and Jürgen Scheuerle for their permission to use this material here. Reduction of Symplectic Manifolds. Most readers of this volume presumably know how symplectic reduction goes: given a hamiltonian action of a Lie group on a symplectic manifold, one divides a level set of a momentum map by the action of a suitable subgroup to form a new symplectic manifold. Before the division step, one has a manifold (possibly