Tudor S. Ratiu

Nonlinear stability of fluid and plasma equilibria

The Liapunov method for establishing stability has been used in a variety of fluid and plasma problems. For nondissipative systems, this stability method is related to well-known energy principles. A development of the Liapunov method for Hamiltonian systems due to Arnold uses the energy plus other conserved quantities, together with second variations and convexity estimates, to establish stability. For Hamiltonian systems, a useful class of these conserved quantities consists of the Casimir functionals, which Poisson-commute with all functionals of the given dynamical variables. Such conserved quantities, when added to the energy, help to provide convexity estimates bounding the growth of perturbations. These estimates enable one to prove nonlinear stability, whereas the commonly used second variation or spectral arguments only prove linearized stability. When combined with recent advances in the Hamiltonian structure of fluid and plasma systems, this convexity method proves to be widely and easily applicable. This paper obtains new nonlinear stability criteria for equilibria for MHD, multifluid plasmas and the Maxwell-Vlasov equations in two and three dimensions. Related systems, such as multilayer quasigeostrophic flow, adiabatic flow and the Poisson-Vlasov equation are also treated. Other related systems, such as stratified flow and reduced magnetohydrodynamic equilibria are mentioned where appropriate, but are treated in detail in other publications.

The Hamiltonian structure for dynamic free boundary problems

Hamiltonian structures for 2- or 3-dimensional incompressible flows with a free boundary are determined which generalize a previous structure of Zakharov for irrotational flow. Our Poisson bracket is determined using the method of Arnold, namely reduction from canonical variables in the Lagrangian (material) description. Using this bracket, the Hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated. The structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained. In the case of two-dimensional flows, the vorticity bracket is determined and the generalized enstrophy is shown to be a Casimir function. This investigation also clears up some confusion in the literature concerning the vorticity bracket, even for fixed boundary flows.

The differentiable structure of three remarkable diffeomorphism groups

The goal of this paper is to find the Lie groups for three well-known Lie algebras: the globally Hamiltonian vector fields, the infinitesimal quantomorphisms and the homogeneous real-valued functions of degree one on the cotangent bundle minus the zero section. In all our considerations the underlying manifold is assumed to be compact without boundary. Before explaining and motivating our results, a very brief review of the Lie group structure of diffeomorphism groups is in order ([9, 10, 18]). If M, N are compact, boundaryless, finite-dimensional manifolds, rN: TN--,N, the tangent bundle of N, an HS-map, s>(dimM)/2, from M to N has by definition all derivatives of order (dim M)/2. Then the space IP(M, N) of all such maps is a Hilbert manifold whose tangent space at every point is the Hilbert space T I. H~(M, N) = {gelid(M, TN) I ~:N ~ g =f} If E---, M is a vector bundle, the same construction works for all HS-sections HS(E) of E, s > (dim M)/2. If N = IR, HS(M, N) will be denoted by C~(M, IR). Let ~ + I ( M ) denote the diffeomorphisms of M of Sobolev class H s+l, i.e. r t ~ + l ( M ) if and only if t/ is bijective and t/, t ] l : M---,M are of class H ~+1. ~s+l(M) is a topological group, and since it is open in Hs+I(M,M), it is also a Hilbert manifold. Right multiplication R~: @~+ ~(M) ~ + I(M), R,(~) = ~ o t/ is C ~ for each t / ~ Y +~(M) and if t/E@ s+k+l(M), left multiplication L , : ~ + t ( M ) ~ + I ( M ) , L~(~)=t/o~ is C k. The inversion map t/~--~t/-1 in ~ + t(M) is only continuous. The tangent maps of R, and L, at e, the identity of ~ + a ( M ) , are given by Tr TeL,(X)=T~oX, where X~Yys+I(M) =HS+I(TM), the set of all H~+l-vector fields on M. The tangent space at e, Tr coincides with Y~+~(M), which is a Sobolev space. The bracket [X, Y] of X, y ~ f s + I(M ) is however only of class H ~ (one derivative is lost). The usual bracket of vector fields is the Lie algebra bracket of 5F ~+ X(M), i.e. if J(,

The Lie algebraic interpretation of the complete integrability of the Rosochatius system

The present note answers a question posed by A.G. Reyman [5] as to the Lie algebraic reasons of the complete integrability of a system studied by E. Rosochatius [6].

Discrete rigid body dynamics and optimal control

We analyze an alternative formulation of the rigid body equations, their relationship with the discrete rigid body equations of Moser-Veselov (1991) and their formulation as an optimal control problem. In addition we discuss a general class of discrete optimal control problems.

Canonical maps between semidirect products with applications to elasticity and superfluids

It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.