Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

Bracken, Paul

doi:10.1007/s10773-005-1489-z

Cited by 4 publications

(8 citation statements)

References 15 publications

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“…We believe that it should be possible to derive a similar theorem in other nonlinear systems. Elliptic compacton solutions also exist in some related equations [24,25]. Our exact solutions and related results should be useful in a wide variety of physical systems ranging from twisting motion of base pairs in DNA models [4] to elasticity in rods and plates [6,7].…”

Section: General Remarksmentioning

confidence: 91%

Exact elliptic compactons in generalized Korteweg–De Vries equations

2006

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Using the action principle, and assuming a solitary wave of the generic form u(x, t) = AZ (β(x + q(t)) Periodicals, Inc. Complexity 11: 30-34, 2006 Key Words: action principle; nonlinear field equation; compacton solutions I n a variety of physical contexts one finds nonlinear field equations for which a wide class of solitary wave solutions can exist. However, in many cases one is not able to obtain the solution in a closed form and thus it is not very easy to study the stability of such solutions. In this article we derive a general theorem relating the energy, momentum, and velocity of any solitary wave of a generalized Korteweg-De Vries (KdV) equation of Cooper et al. [1]. The important point is that this particular generalization of the KdV equation is derivable Correspondence to : Avadh Saxena (e-mail: avadh@lanl.gov) from an action principle. Using the theorem, we are able to relate the amplitude, width, and velocity of any of the solitary wave solution even if such a solution is not known in a closed form and also to study its stability. Second, we obtain a two-parameter family of solutions to these equations which include elliptic and hyper-elliptic compacton solutions. For this general family we explicitly verify the theorem as well as the stability criteria.Compactons were discovered originally in an extension

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Section: General Remarksmentioning

confidence: 91%

Exact elliptic compactons in generalized Korteweg–De Vries equations

2006

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“…[3,11,18]). Substitution of the latter into (6) suggests that the electric field intensity can be written as E = −A t − grad φ, where φ represents the electric scalar potential.…”

Section: Classical Approachmentioning

confidence: 99%

“…This framework might especially be of interest in cases where it is desirable to consider a set of symmetric Maxwell equations, i.e., in cases where J m = 0 and div B = 0. As argued in [3], Maxwell's equations can be partitioned into kinematical equations, Faraday's law (6) and (9), and dynamical equations, the Ampère-Maxwell law (7) and Gauss' law (8). On the other hand, storage of energy is associated with the dynamics of a system.…”

Section: Final Remarks and Outlookmentioning

confidence: 99%

“…Furthermore, the construction of an electromagnetic mixed-potential also gives rise to some alternative variational principles that imply the existence of a family of novel Lagrangian functionals. In contrast to the existing methods in mathematical physics, these Lagrangian functionals explicitly yield both Maxwell's curl equations explicitly without the usual restriction to systems with infinite spatial domain, where the dynamic variables go to zero for the spatial variables tending to infinity, and without imposing the usual assumption that the energy exchange through the boundary is zero [3,22]. This leads to the definition of an infinite-dimensional Lagrangian boundary control system, as originally introduced for mechanical systems in [6] (see also [21] for a summary and further developments on the topic).…”

Section: Contribution and Outlinementioning

confidence: 99%

“…The subscript notation (•) u denotes partial differentiation with respect to u. In case of a topologically complete 3 network the mixed-potential function takes the form…”

Section: Introductionmentioning

confidence: 99%

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Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic fields

Jeltsema¹,

Schaft²

2007

J. Phys. A: Math. Theor.

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This paper describes an electromagnetic field analogue of the classical BraytonMoser formulation. It is shown that Maxwell's curl equations constitute a pseudo-gradient system with respect to a single electromagnetic mixedpotential functional and a metric defined by the constitutive relations of the fields. Besides its use for the generation of power-based Lyapunov functionals for stability analysis and Poynting-like flow balances, the electromagnetic mixed-potential formulation suggests a family of alternative variational principles. This yields a novel Lagrangian boundary control system formulation admitting nonzero energy flow through the boundary. The corresponding symplectic Hamiltonian system is still associated with the total electromagnetic field energy.

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Non-Abelian gauge theory from the Poisson bracket

Chen

Nyeo

2018

Int. J. Mod. Phys. A

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The Hamiltonian of a nonrelativistic particle coupled to non-Abelian gauge fields is defined to construct a non-Abelian gauge theory. The Hamiltonian which includes isospin as a dynamical variable dictates the dynamics of the particle and isospin according to the Poisson bracket that incorporates the Lie algebraic structure of isospin. The generalized Poisson bracket allows us to derive Wong’s equations, which describe the dynamics of isospin, and the homogeneous (sourceless) equations for non-Abelian gauge fields by following Feynman’s proof of the homogeneous Maxwell equations.It is shown that the derivation of the homogeneous equations for non-Abelian gauge fields using the generalized Poisson bracket does not require that Wong’s equations be defined in the time-axial gauge, which was used with the commutation relation. The homogeneous equations derived by using the commutation relation are not Galilean and Lorentz invariant. However, by using the generalized Poisson bracket, it can be shown that the homogeneous equations are not only Galilean and Lorentz invariant but also gauge independent. In addition, the quantum ordering ambiguity that arises from using the commutation relation can be avoided when using the Poisson bracket.From the homogeneous equations, which define the “electric field” and “magnetic field” in terms of non-Abelian gauge fields, we construct the gauge and Lorentz invariant Lagrangian density and derive the inhomogeneous equations that describe the interaction of non-Abelian gauge fields with a particle.

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Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

Abstract: The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the

Cited by 4 publications

References 15 publications

Exact elliptic compactons in generalized Korteweg–De Vries equations

Exact elliptic compactons in generalized Korteweg–De Vries equations

Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic fields

Non-Abelian gauge theory from the Poisson bracket

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