On some applications of Galilean electrodynamics of moving bodies

Montigny, M. de; Rousseaux, Germain

doi:10.1119/1.2772289

Cited by 26 publications

(25 citation statements)

References 44 publications

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“…is Galilean invariant). A gauge choice which achieves this, first suggested to us by Brandenburg (private communication), is to choose φ= v · A , which leads to the induction equation in the form It turns out that there are other good reasons for this choice of gauge – most notably that this in fact represents the correct low‐speed ( v ≪ c ) and magnetically dominated ( E ≪ cB ) limit for electromagnetism (de Montigny & Rousseaux 2007). Written in tensor notation, can be expressed by where ε ijk is the Levi‐Civita permutation tensor and repeated indices imply a summation.…”

Section: A Consistent Formulation Of Spmhd Using the Vector Potentialmentioning

confidence: 99%

Smoothed Particle Magnetohydrodynamics - IV. Using the vector potential

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2010

Monthly Notices of the Royal Astronomical Society

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In this paper we investigate the use of the vector potential as a means of maintaining the divergence constraint in the numerical solution of the equations of Magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. We derive a self-consistent formulation of the equations of motion using a variational principle that is constrained by the numerical formulation of both the induction equation and the curl operator used to obtain the magnetic field, which guarantees exact and simultaneous conservation of momentum, energy and entropy in the numerical scheme. This leads to a novel formulation of the MHD force term, unique to the vector potential, which differs from previous formulations. We also demonstrate how dissipative terms can be correctly formulated for the vector potential such that the contribution to the entropy is positive definite and the total energy is conserved. On a standard suite of numerical tests in one, two and three dimensions we find firstly that the consistent formulation of the vector potential equations is unstable to the well-known SPH tensile instability, even more so than in the standard Smoothed Particle Magnetohydro-dynamics (SPMHD) formulation where the magnetic field is evolved directly. Furthermore we find that, whilst a hybrid approach based on the vector potential evolution equation coupled with a standard force term gives good results for one and two dimensional problems (where dA z /dt = 0), such an approach suffers from numerical instability in three dimensions related to the unconstrained evolution of vector potential components. We conclude that use of the vector potential is not a viable approach for Smoothed Particle Magnetohydrodynamics.

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Section: A Consistent Formulation Of Spmhd Using the Vector Potentialmentioning

confidence: 99%

Smoothed Particle Magnetohydrodynamics - IV. Using the vector potential

Price

2010

Monthly Notices of the Royal Astronomical Society

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“…Next, an assumption on the relative magnitude of the remaining terms must be added in order to drop the ones which break the Galilean covariance. It is easy to see that the magnetic limit corresponds to the assumption E m ∼ vB m cB m (see [13][14][15][16]). Hence, the Galilean magnetic constitutive equations write:…”

Section: Relativistic Electrodynamics Of Moving Media -mentioning

confidence: 99%

“…whereas the electric limit corresponds to cB e ∼ vE e / c E e (see [13][14][15][16]) with the following Galilean electric constitutive equations:…”

Section: Relativistic Electrodynamics Of Moving Media -mentioning

confidence: 99%

On the electrodynamics of Minkowski at low velocities

Rousseaux

2008

Europhys. Lett.

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“…Equations (3.4) and (3.5) provide the consistently pre-relativistic transformation rules for the fields guaranteeing the Galilean invariance of the Lorentz force (3.1). It is worth noting that these equations actually coincide with the nonrelativistic 'magnetic' limit of the Lorentz transformations for the fields, as defined in [19].…”

Section: Pre-relativistic Requirement Of Lorentz Force Invariancementioning

confidence: 79%

On the Galilean non-invariance of classical electromagnetism

2009

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When asked to explain the Galilean non-invariance of classical electromagnetism on the basis of pre-relativistic considerations alone, students-and sometimes their teachers too-may face an impasse. Indeed, they often argue that a pre-relativistic physicist could most obviously have provided the explanation 'at a glance', on the basis of the presence of a parameter c with the dimensions of a velocity in Maxwell's equations, being well aware of the fact that any velocity is non-invariant in Galilean relativity. This 'obvious' answer, however popular, is not correct due to the actual observer-invariance of the Maxwell parameter c in pre-relativistic physics too. A pre-relativistic physicist would therefore have needed a different explanation. Playing the role of this physicist, we pedagogically show how a proof of the Galilean non-invariance of classical electromagnetism can be obtained, resting on simple pre-relativistic considerations alone.

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On some applications of Galilean electrodynamics of moving bodies

Cited by 26 publications

References 44 publications

Smoothed Particle Magnetohydrodynamics - IV. Using the vector potential

Smoothed Particle Magnetohydrodynamics - IV. Using the vector potential

On the electrodynamics of Minkowski at low velocities

On the Galilean non-invariance of classical electromagnetism

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