In this chapter, we extend the theory of reduction of Hamiltonian systems with symmetry to include systems with a discrete symmetry group acting symplectically. The exposition here is based on the work of Harnad, Hurtubise and Marsden [1991].

For antisymplectic symmetries such as reversibility, this question has been considered by Meyer [1981] and Wan [1990]. However, in this chapter we are concerned with symplectic symmetries. Antisymplectic symmetries are typified by time reversal symmetry, while symplectic symmetries are typified by spatial discrete symmetries of systems like reflection symmetry. Often these are obtained by taking the cotangent lift of a discrete symmetry of configuration space.

There are two main motivations for the study of discrete symmetries. The first is the theory of bifurcation of relative equilibria in mechanical systems with symmetry. The rotating liquid drop is a system with a symmetric relative equilibrium that bifurcates via a discrete symmetry. An initially circular drop (with symmetry group S1) that is rotating rigidly in the plane with constant angular velocity Ω, radius r, and with surface tension τ, is stable if r3Ω2 < 12τ (this is proved by the energy-Casimir or energy-momentum method). Another relative equilibrium (a rigidly rotating solution in this example) branches from this circular solution at the critical point r3Ω2 = 12τ. The new solution has the spatial symmetry of an ellipse; that is, it has the symmetry ℤ2 × ℤ2 (or equivalently, the dihedral group D2).

In this chapter we discuss the cotangent bundle reduction theorem. Versions of this are already given in Smale [1970], but primarily for the abelian case. This was amplified in the work of Satzer [1977] and motivated by this, was extended to the nonabelian case in Abraham and Marsden [1978]. An important formulation of this was given by Kummer [1981] in terms of connections. Building on this, the “bundle picture” was developed by Montgomery, Marsden and Ratiu [1984] and Montgomery [1986].

From the symplectic viewpoint, the principal result is that the reduction of a cotangent bundle T*Q at µ ∈ g* is a bundle over T*(Q/G) with fiber the coadjoint orbit through µ. Here, S = Q/G is called shape space. From the Poisson viewpoint, this reads: (T*Q)/G is a g*-bundle over T*(Q/G), or a Lie-Poisson bundle over the cotangent bundle of shape space. We describe the geometry of this reduction using the mechanical connection and explicate the reduced symplectic structure and the reduced Hamiltonian for simple mechanical systems.

Mechanical G-systems

By a symplectic (resp. Poisson) G-system we mean a symplectic (resp. Poisson) manifold (P, Ω) together with the symplectic action of a Lie group G on P, an equivariant momentum map J : P → g* and a G-invariant Hamiltonian H : P → ℝ.

Following terminology of Smale [1970], we refer to the following special case of a symplectic G-system as a simple mechanical G-system.

In this chapter, we study some examples of bifurcations in the Hamiltonian context. A lot of the ideas from the previous chapters come into this discussion, and links with new ones get established, such as connections with chaotic dynamics and solution spaces in relativistic field theories. Our discussion will be by no means complete; it will focus on certain results of personal interest and results that fit in with the rest of the chapters. Some additional information on bifurcation theory in the Hamiltonian context may be found in the references cited below and in Abraham and Marsden [1978], Arnold [1978], Meyer and Hall [1991] and the references therein.

Some Introductory Examples

Bifurcation theory deals with the changes in the phase portrait structure of a given dynamical system as parameters are varied. One usually begins by focussing on the simplest features of the phase portrait, such as equilibrium points, relative equilibria, periodic orbits, relative periodic orbits, homoclinic orbits, etc., and studies how they change in number and stability characteristics as the system parameters are changed. Often these changes lead to new structures, such as more equilibria, periodic orbits, tori, or chaotic solutions, and the way in which stability or instability is transfered to these new structures from the old ones is of interest.

For conservative mechanical systems with symmetry, it is of interest to develop numerical schemes that preserve this symmetry, so that the associated conserved quantities are preserved exactly by the integration process. One would also like the algorithm to preserve either the Hamiltonian or the symplectic structure — one cannot expect to do both in general, as we shall show below. There is some evidence (such as reported by Chanell and Scovel [1990] and Marsden et al. [1991]) that these mechanical integrators perform especially well for long time integrations, in which chaotic dynamics can be expected. It is well known that, in general, may standard algorithms can introduce spurious effects (such as nonexistent chaos) in long integration runs; see, for example, Reinhall, Caughey, and Storti [1989]. We use the general term mechanical integrator for an algorithm that respects one or more of the fundamental properties of being symplectic, preserving energy, or preserving the momentum map.

Definitions and Examples

By an algorithm on a phase space P we mean a collection of maps Fτ: P → P (depending smoothly, say, on τ ∈ ℝ for small τ and z ∈ P). Sometimes we write zk+1 = Fτ(zk) for the algorithm and we write Δt or h for the step size τ.

In setting down a general continuum mechanics description of finite deformation processes in metal crystals that takes into account both lattice straining and gross crystallographic slip, it is useful to begin with an assessment of the minimum physical scale at which such a description has meaning. The following discussion, based upon similar discussions in Havner (1973a,b; 1982a), is pertinent to the determination of that minimum scale.

As seen by an observer resolving distances to 10−3 mm, the deformation of a crystal grain (of typical dimensions 10−3−10−2 cm within a finitely strained polycrystalline metal) may be considered relatively smooth. At this level of observation, which for convenience we shall call microscopic, one can just distinguish between slip lines on crystal faces after extensive distortion of a specimen. In contrast, a submicroscopic observer resolving distances to 10−5 mm (the order of 100 atomic spacings) is aware of highly discontinuous displacements within crystals. The microscopic observer's slip lines appear to the submicroscopic observer as slip bands of order 10−4 mm thickness, containing numerous glide lamellae between which amounts of slip as great as 103 lattice spacings have occurred, as first reported by Heidenreich (1949); hence a continuum perspective at this second level would seem untenable. Accordingly, we adopt a continuum model in which a material “point” has physical dimensions of order 10−3 mm. This is greater than 103 lattice spacings yet at least an order of magnitude smaller than typical grain sizes in polycrystalline metals.

In this chapter we return to the general theoretical framework of Chapter 3 and extend it to the analysis of characteristics of overall response of macroscopically uniform polycrystalline solids. The objective is the presentation of a rigorous theoretical connection between single-crystal elastoplasticity and macroscopic crystalline aggregate behavior. The development is based upon the original analysis of Hill (1972) and other basic contributions in Hill & Rice (1973), Havner (1974, 1982a, 1986), and Hill (1984, 1985). Central to an understanding of the crystal-to-aggregate transition is the well-known “averaging theorem” introduced by Bishop & Hill (1951a) but only given its final form and initial proof at finite strain in Hill's (1972) seminal work.

Crystalline Aggregate Model: The Averaging Theorem

At the beginning of Chapter 3, the scale of a crystal material point in a continuum model was defined to have linear dimension of order 10−3 mm: greater than 103 lattice spacings but at least an order of magnitude smaller than normal grain sizes in polycrystalline metals. Consider now the choice of physical size of a representative “macroelement” that defines a continuum point at the level of ordinary stress and strain analysis (that is, in structural and mechanical components or materials-forming operations.)

The wall thickness of thin-walled metal tubes used in combined stress tests (say, axial loading and torsion) often is in the range 1−2 mm and 10 to 30 grains (see, for example, Mair & Pugh (1964) or Ronay (1968)).

Turning from the rigorous theoretical analysis of Chapter 6 to the subject of (and literature on) the calculation of approximate polycrystalline aggregate models at finite strain, one can identify three prominent themes: the prediction of (i) macroscopic axial-stress–strain curves, (ii) macroscopic yield loci, and (iii) the evolution of textures (that is, the development of preferred crystal orientations in initially statistically isotropic aggregates). The topic of polycrystal calculations is vast and complex, warranting a monograph on its own (and by other hands). In this closing chapter of the present work I primarily shall review selected papers (acknowledging others) from among those contributions that are particularly significant or noteworthy in the more than 50 years' history of the subject.

The Classic Theories of Taylor, Bishop, and Hill

Near the beginning of G. I. Taylor's (1938a) May Lecture to the Institute of Metals is the following splendid sentence. “I must begin by making the confession that I am not a metallurgist; I may say, however, that I have had the advantage of help from, and collaboration with, members of your Institute, whose names are a sure guarantee that the metals I have used were all right, even if my theories about them are all wrong.” More than anything else this statement reflects Taylor's irrepressible humor, for of course his theories were not “all wrong.”