0 A pr 1 99 8 Invariants and Labels in Lie – Poisson Systems

Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket obtained is not of the canonical type. Specifically, we give two examples that give rise to brackets of the noncanonical Lie–Poisson form: the rigid body and the two-dimensional ideal fluid. From these simple cases, we then use the semidirect product extension of algebras to describe more complex physical systems. The Casimir invariants in these systems are examined, and some are shown to be linked to the recovery of information about the configuration of the system. We discuss a case in which the extension is not a semidirect product, namely compressible reduced MHD, and find for this case that the Casimir invariants lend partial information about the configuration of the system. e-mail: [email protected], Tel.: (512)471-6121, Fax: (512)471-6715 e-mail: [email protected] 1 This paper explores the Casimir invariants of Lie–Poisson brackets, which generate the dynamics of some discrete and continuous Hamiltonian systems. Lie–Poisson brackets are a type of noncanonical Poisson bracket and are ubiquitous in the reduction of canonical Hamiltonian systems with symmetry. Casimir invariants are constants of motion for all Hamiltonians; they are associated with the degeneracy of noncanonical Poisson brackets. Finitedimensional examples of systems described by Lie–Poisson brackets include the heavy top and the moment reduction of the Kida vortex, while infinite-dimensional examples include the 2–D ideal fluid, reduced magnetohydrodynamics (MHD), and the 1–D Vlasov equation. (See Ref. 1 and references therein for a full review.) The Casimir invariants determine the manifold on which the system is kinematically constrained to evolve. Understanding the nature of these constraints is thus of paramount importance. In Section I we examine specific Lie–Poisson brackets, namely, those that arise from the reduction to Eulerian variables of a Lagrangian system with relabeling symmetry. We make use of two prototypical examples, the rigid body (finite-dimensional) and the 2–D ideal fluid (infinite-dimensional), and we interpret their Casimir invariants. This is done to motivate the introduction of such brackets and to show their physical relevance. In Section II we turn to building Lie–Poisson brackets directly from Lie algebras by the procedure of extension. We introduce the semidirect product extension and illustrate it with two physical examples: the heavy top and low-beta reduced MHD. Finally in Section III we look at a nonsemidirect example, compressible reduced MHD, and discuss work in progress. I. LIE–POISSON BRACKETS AND REDUCTION For our purposes, a reduction is a mapping of the dynamical variables of a system to a smaller set of variables, such that the transformed Hamiltonian and bracket depend only on the smaller set of variables. (See for example Ref. 2 for a detailed treatment.) The simplest example of a reduction is the case in which a cyclic variable is eliminated, but more generally a reduction exists as a consequence of an underlying symmetry of the system. We present two examples of reduction. A. Reduction of the Free Rigid Body The Hamiltonian for the free rigid body is an unwieldy function of three Euler angles φ, ψ, θ and their conjugate momenta pφ, pψ, pθ. The motion is described by Hamilton’s equations using the canonical bracket {f , g}can = ∂f ∂φ ∂g ∂pφ − ∂g ∂φ ∂f ∂pφ + ∂f ∂ψ ∂g ∂pψ − ∂g ∂ψ ∂f ∂pψ + ∂f ∂θ ∂g ∂pθ − ∂g ∂θ ∂f ∂pθ . Here we have 3 degrees of freedom (6 coordinates), the configuration space is the rotation group SO(3), and the phase space is its contangent bundle T SO(3). A reduction is possible for this system. In terms of angular momenta li about the principal axes, we have H(φ, ψ, θ, pφ, pψ, pθ) −→ H(l1, l2, l3) = 3 ∑