Relabeling symmetries in hydrodynamics and magnetohydrodynamics

Padhye, Nikhil; Morrison, Philip

doi:10.2172/226406

Cited by 54 publications

(115 citation statements)

References 11 publications

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“…A = B −1 . It should be emphasized that the transformation of B and its inverse is not connected at this point to the dynamics in any way, nor is it related to symmetry as in [16][17][18]. This kind of label change does not usually appear in traditional finite-dimensional Hamiltonian theory, since it would amount to a time-dependent change of the label i of, e.g., a canonical coordinate q i (t).…”

Section: Lagrangian Stability a General Hamiltonian Form Relabementioning

confidence: 98%

“…The phase space of this setup is sometimes denoted by T * Q, where Q is the set of smooth invertible mappings of the spatial domain, indicated here by q, and T * Q denotes the space (cotangent bundle) with coordinates q together with their conjugate momenta π. Because the infinitedimensional geometry implied by T * Q is backed by meager mathematical rigor, the language of Lagrange [14] and Newcomb [15] will be used here, except because general curvilinear coordinates may be employed indices will be placed as in [3,16,17] indicating their tensorial character, viz., q → q i and π → π i , where i = 1, 2, 3.…”

Section: Lagrangian Stability a General Hamiltonian Form Relabementioning

confidence: 99%

See 1 more Smart Citation

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

2013

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Stability conditions of magnetized plasma flows are obtained by exploiting the Hamiltonian structure of the magnetohydrodynamics (MHD) equations and, in particular, by using three kinds of energy principles. First, the Lagrangian variable energy principle is described and sufficient stability conditions are presented. Next, plasma flows are described in terms of Eulerian variables and the noncanonical Hamiltonian formulation of MHD is exploited. For symmetric equilibria, the energy-Casimir principle is expanded to second order and sufficient conditions for stability to symmetric perturbation are obtained. Then, dynamically accessible variations, i.e. variations that explicitly preserve invariants of the system, are introduced and the respective energy principle is considered. General criteria for stability are obtained, along with comparisons between the three different approaches.

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Section: Lagrangian Stability a General Hamiltonian Form Relabementioning

confidence: 98%

Section: Lagrangian Stability a General Hamiltonian Form Relabementioning

confidence: 99%

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

2013

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“…Other relevant references are [20,23,24]. We will pretend we don't know the answer and proceed as if we were building a theory from the ground up.…”

Section: Mhd Actionmentioning

confidence: 99%

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

Morrison

Eliasson²,

Shukła³

2009

AIP Conference Proceedings

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Abstract.A general discussion of Hamiltonian and action principle formulations for fliud and plasma models is given. A procedure, based on Hamilton's principle of mechanics but adapted for continua, for the construction of action principles for fliud and kinetic models is given. The transformation from action principles in terms of the Lagrangian variable description to the Eulerian variable description in terms of noncanonical Poisson brackets is described. Two examples are developed: ideal MHD and Braginskii's fliud model with gyroviscosity.

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“…Existence itself of the ideal-hydrodynamic solutions in the form of quasi-one-dimensional vortex structures (vortex filaments) filling just a small part of the total fluid bulk is closely connected with the freezing-in property of the vortex lines [1,2,3,4,5,6,7,8,9,10,11,12,13]. Mathematically this property is expressed by the special form of the equation of motion for the divergence-free vorticity field Ω(r, t) = curl v(r, t),…”

Section: Long-scale Local Approximationmentioning

confidence: 99%

Applicability of the hodograph method for the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary

Ruban

2003

Phys. Rev. E

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Hamiltonian dynamics of a thin vortex filament in ideal incompressible fluid near a flat fixed boundary is considered at the conditions that at any point of the curve determining shape of the filament the angle between tangent vector and the boundary plane is small, also the distance from a point on the curve to the plane is small in comparison with the curvature radius. The dynamics is shown to be effectively described by a nonlinear system of two (1+1)-dimensional partial differential equations. The hodograph transformation reduces that system to a single linear differential equation of the second order with separable variables. Simple solutions of the linear equation are investigated at real values of spectral parameter λ when the filament projection on the boundary plane has shape of a two-branch spiral or a smoothed angle, depending on the sign of λ.

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Relabeling symmetries in hydrodynamics and magnetohydrodynamics

Cited by 54 publications

References 11 publications

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

Applicability of the hodograph method for the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary

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