Solutions of the Wheeler-Feynman equations with discontinuous velocities

Souza, Daniel Câmara de; Luca, Jayme De

doi:10.1063/1.4905201

Cited by 17 publications

(13 citation statements)

References 20 publications

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“…The present author and collaborators have studied solutions with velocity discontinuities for the two-body problem in Refs. [17,18,29]. In Ref.…”

Section: Existence and Uniqueness Resultsmentioning

confidence: 99%

“…The method [18] still could not deal with velocity discontinuities and had to adjust the past and the future boundary segments to yield C 1 solutions. (vi) Finally, an example of the effectiveness of the variational principle when compared to the neutral differential-delay equation formulation is our first numerical method capable of dealing with velocity discontinuities for boundaries having the shortest time separation between boundary segments across the lightcone [29]. A property of the shortest-length boundaries setup of Ref.…”

Section: B Solution Based Approaches To the Electromagnetic Problemmentioning

confidence: 99%

“…[29] is that one particle's retarded position falls in the past boundary segment while the other particle's advanced position falls in the future boundary segment, thus reducing the functional equation to a shooting problem for a nonautonomous ODE. The method [29] starts from boundary segments that include velocity discontinuities and marches an ODE integrator from the initial point to the endpoint, stopping to enforce the Weierstrass-Erdmann conditions at each breaking point. Solutions with discontinuous velocities are exhibited in Ref.…”

Section: B Solution Based Approaches To the Electromagnetic Problemmentioning

confidence: 99%

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Equations of motion for variational electrodynamics

Luca

2016

Journal of Differential Equations

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We extend the variational problem of Wheeler-Feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative (velocities) of bounded variation. We show here that the Gateaux derivative of the generalized functional defines two partial Lagrangians for variations in our generalized local space, one for each particle. We prove that the critical-point conditions of the generalized variational problem are: (i) the Euler-Lagrange equations must hold Lebesgue-almost-everywhere and (ii) the momentum of each partial Lagrangian and the Legendre transform of each partial Lagrangian must be absolutely continuous functions, generalizing the Weierstrass-Erdmann conditions.

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“…The present author and collaborators have studied solutions with velocity discontinuities for the two-body problem in Refs. [17,18,29]. In Ref.…”

Section: Existence and Uniqueness Resultsmentioning

confidence: 99%

Section: B Solution Based Approaches To the Electromagnetic Problemmentioning

confidence: 99%

Section: B Solution Based Approaches To the Electromagnetic Problemmentioning

confidence: 99%

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Equations of motion for variational electrodynamics

Luca

2016

Journal of Differential Equations

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“…• Unperturbed electrodynamics is not a semiflow on C 2 (R) and must be studied as a boundary-value problem [6][7][8].…”

Section: A Significance Of the Problem And Contentsmentioning

confidence: 99%

“…In view of Lemma 2.1, action (8) is well defined in the domain of trajectory pairs where the denominators of (8) are Lebesgue integrable, i.e.,…”

Section: A Perturbed Functionalmentioning

confidence: 99%

Lorentz-equivariant flow with four delays of neutral type

Luca¹

2021

Preprint

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The electromagnetic two-body problem must be solved as a variational boundaryvalue problem because it is not a semiflow on C 2 (R). In order to allow forward integration by the method of steps, we generalize electrodynamics by adding another time-reversible interaction along lightcones. The equations of motion are time-reversible and define a flow on C 2 (R) with four state-dependent delays of neutral type and nonlinear gyroscopic terms. The flow includes an inversion boundary layer where attraction turns into repulsion. Moreover, the forward propagation of velocity discontinuities requires two additional constraints; the energetic Weierstrass-Erdmann conditions, which define the boundary layer neighborhood where velocity denominators become small. In order to display the boundary layers in the simplest form, we discuss the motion restricted to a straight line by the initial condition.

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Lorentz-equivariant flow with four delays of neutral type

Luca

2022

Journal of Differential Equations

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Solutions of the Wheeler-Feynman equations with discontinuous velocities

Cited by 17 publications

References 20 publications

Equations of motion for variational electrodynamics

Equations of motion for variational electrodynamics

Lorentz-equivariant flow with four delays of neutral type

Lorentz-equivariant flow with four delays of neutral type

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