Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Wave propagation and group velocity

The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable whenmore » nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods. {copyright} {ital 1998} {ital The American Physical Society}« less

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Hamiltonian description of the ideal fluid

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule.

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Lectures on Mechanics

Criteria useful for the investigation of the stability of a system of charged particles are derived from the Boltzmann equation in the small m/e limit. These criteria are obtained from the examination of the variation of the energy due to a perturbation, when subject to the general constraint that all regular, time‐independent constants of the motion (including the energy) have their equilibrium values.The first‐order variation of the energy vanishes trivially, and the second‐order variation yields a quadratic form in the displacement variable ξ (which may be introduced because of the well‐known properties of this limit). The positive‐definiteness of this form is a sufficient condition for stability.Several theorems are stated comparing stability under the present theory with that under conventional hydromagnetic fluid theories where heat flow along magnetic lines of force is neglected. Generalizations can be made to systems where the effect of collisions is included.

On the Stability of Plasma in Static Equilibrium

A previously derived expression [Phys Rev A 40, 3898 (1989)] for the energy of arbitrary perturbations about arbitrary Vlasov–Maxwell equilibria is transformed into a very compact form The new form is also obtained by a canonical transformation method for solving Vlasov’s equation, which is based on Lie group theory This method is simpler than the one used before and provides better physical insight Finally, a procedure is presented for determining the existence of negative‐energy modes In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed

The free energy of Maxwell--Vlasov equilibria

Expressions for the energy content of one‐dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well‐known dielectric energy, ED, is compared with the exact plasma free energy expression, δ2F, that is conserved by the Vlasov–Poisson system [Phys. Rev. A 40, 3898 (1989) and Phys. Fluids B 2, 1105 (1990)]. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbation of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes δ2F=ED. In the case of unstable and corresponding damped modes it is seen that δ2F≠ED; in fact δ2F≡0. This fail...

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Dielectric energy versus plasma energy, and Hamiltonian action‐angle variables for the Vlasov equation

expressions for the energy and energy flux densities were determined. Here the full energy-momentum tensor and the angular momentum tensor are obtained via Noether's theorem. The appropriate symmetries of these tensors are shown and the local conservation laws are proven. These results are of importance for applications; e.g., knowledge of the total angular momentum combined with the energy can lead to an "energy" principle for linear stability analysis. The tensors are obtained in the usual way by first deter­ mining the general expression for an arbitrary variation of the total· Lagrangian density in normal position space x. We note, however, that there is a slight complication be­ cause the "particle" part of the Lagrangian is primarily defined on an extended space Y=(Y"Y2) where y, is iden­ tical to x and Y2 is an additional coordinate that is needed in order to describe guiding centers. By means of transla­ tional invariance in x space and time, and rotational in­ variance in x space· the canonical tensors are obtained. These tensors are not gauge invariant, but each can be split into a divergence-free gauge-invariant part and a divergence-free non-gauge-invariant part. The gauge­ invariant energy-momentum tensor turns out to be sym­ metric in its spatial components. This is shown to follow from its relationship to the gauge-iI,lVariant part of the an­ gular momentum tensor. All of these expressions are also applicable to relativistic theories. Two applications are considered: first we treat the Maxwell-Vlasov equations and show that the gauge­ invariant parts of our tensors reduce to the usual well­ known expressions. Following this we treat the Maxwell­ kinetic guiding-center theory based on Littlejohn's guiding-center equations of motion.4

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Local conservation laws for the Maxwell-Vlasov and collisionless kinetic guiding-center theories.

A modified Hamilton–Jacobi formalism is introduced as a tool to obtain the energy‐momentum and angular‐momentum tensors for any kind of nonlinear or linearized Maxwell‐collisionless kinetic theories. The emphasis is on linearized theories, for which these tensors are derived for the first time. The kinetic theories treated—which need not be the same for all particle species in a plasma—are the Vlasov and kinetic guiding center theories. The Hamiltonian for the guiding center motion is taken in the form resulting from Dirac’s constraint theory for nonstandard Lagrangian systems. As an example of the Maxwell‐kinetic guiding center theory, the second‐order energy for a perturbed homogeneous magnetized plasma is calculated with initially vanishing field perturbations. The expression obtained is compared with the corresponding one of Maxwell–Vlasov theory.

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The energy-momentum tensor for the linearized Maxwell-Vlasov and kinetic guiding center theories

Explosive phenomena such as internal disruptions in toroidal discharges and solar flares are difficult to explain in terms of linear instabilities. A plasma approaching a linear stability limit can, however, become nonlinearly and explosively unstable, with noninfinitesimal perturbations even before the marginal state is reached. For such investigations, a nonlinear extension of the usual MHD (magnetohydrodynamic) energy principle is helpful. (This was obtained by Merkel and Schluter, Sitzungsberichted. Bayer. Akad. Wiss., Munich, 1976, No. 7, for Cartesian coordinate systems.) A coordinate system independent Eulerian formulation for the Lagrangian allowing for equilibria with flow and with built‐in conservation laws for mass, magnetic flux, and entropy is developed in this paper which is similar to Newcomb’s Lagrangian method of 1962 [Nucl. Fusion, Suppl., Pt. II, 452 (1962)]. For static equilibria nonlinear stability is completely determined by the potential energy. For a potential energy which contains second‐ and nth order or some more general contributions only, it is shown in full generality that linearly unstable and marginally stable systems are explosively unstable even for infinitesimal perturbations; linearly absolutely stable systems require finite initial perturbations. For equilibria with Abelian symmetries symmetry breaking initial perturbations are needed, which should be observed in numerical simulations. Nonlinear stability is proved for two simple examples, m=0 perturbations of a Bennet Z‐pinch and z‐independent perturbations of a θ pinch. The algebra for treating these cases reduces considerably if symmetries are taken into account from the outset, as suggested by M. N. Rosenbluth (private communication, 1992).

D. Pfirsch

Papers

The free energy of Maxwell--Vlasov equilibria

Dielectric energy versus plasma energy, and Hamiltonian action‐angle variables for the Vlasov equation

Local conservation laws for the Maxwell-Vlasov and collisionless kinetic guiding-center theories.

The energy-momentum tensor for the linearized Maxwell-Vlasov and kinetic guiding center theories

Nonlinear ideal magnetohydrodynamics instabilities