Can the future influence the present?

Driver, R. D.

doi:10.1103/physrevd.19.1098

Cited by 51 publications

(65 citation statements)

References 13 publications

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“…The symmetric solution was actually proved to be unique only up to a small velocity [13], and we managed to go much above the low limit set by Driver in Ref. [13]. In approximately a linear function of (r o /r) at low velocities, but at larger velocities it becomes highly convoluted.…”

Section: Numerical Integration: the Steepest-descent Methodsmentioning

confidence: 63%

“…[13] for sufficiently low velocity orbits (in Appendix 1 we prove this assertion for the arbitrary-mass case). We henceforth refer to a CMF orbit as a twicemonotonic orbit.…”

Section: Outline Of the Methodsmentioning

confidence: 64%

“…In this high-velocity region the functional problem posed on F (r) might not have a unique solution, and for relativistic velocities it is likely that it does not. The symmetric solution was actually proved to be unique only up to a small velocity [13], and we managed to go much above the low limit set by Driver in Ref. [13].…”

Section: Numerical Integration: the Steepest-descent Methodsmentioning

confidence: 72%

“…Now if asymptotic data determine a unique trajectory for low energies, that trajectory is our symmetric solution in this CMF. Of course the symmetric solution is always a possible solution, but the symmetry of the physics in this outbound CMF suggests that it is the only solution for low energies, as the solution has to correspond to the Coulombian solution, which has this property (this would actually be a non-trivial generalization of the work of Driver and Hoag in references [13,14]). With the above in mind, the Hamiltonian we derive here is already the general order-reduced Hamiltonian for low energies.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In general, a neutral-delay equation such as Eqs. (1) and (2) requires an initial function as the initial condition, but for the special case of equations (1) and (2) it was proved in 1979 that for sufficiently low energies the Newtonian initial condition [ x(0) = x o and v(0) = v o ] determines a unique symmetric solution that is globally defined (i.e., that does not runaway at some point) [13,14]. This surprising uniqueness theorem reducing the initial condition from an arbitrary function to two simple real numbers (initial position and velocity) already suggests that the physical phase space could be isomorphic to a two-degree-of-freedom Hamiltonian vector field, at least for low velocities (which is what we find here).…”

Section: Introductionmentioning

confidence: 99%

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Two-degree-of-freedom Hamiltonian for the time-symmetric two-body problem of the relativistic action-at-a-distance electrodynamics

Hollander

Luca

2003

Phys. Rev. E

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We find a two-degree-of-freedom Hamiltonian for the time-symmetric problem of straight line motion of two electrons in direct relativistic interaction. This time-symmetric dynamical system appeared 100 years ago and it was popularized in the 1940's by the work of Wheeler and Feynman in electrodynamics, which was left incomplete due to the lack of a Hamiltonian description. The form of our Hamiltonian is such that the action of a Lorentz transformation is explicitly described by a canonical transformation (with rescaling of the evolution parameter). The method is closed and defines the Hamitonian in implicit form without power expansions. We outline the method with an emphasis on the physics of this complex conservative dynamical system. The Hamiltonian orbits are calculated numerically at low energies using a self-consistent steepest-descent method (a stable numerical method that chooses only the nonrunaway solution). The two-degree-of-freedom Hamiltonian suggests a simple prescription for the canonical quantization of the relativistic twobody problem.

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Section: Numerical Integration: the Steepest-descent Methodsmentioning

confidence: 63%

“…[13] for sufficiently low velocity orbits (in Appendix 1 we prove this assertion for the arbitrary-mass case). We henceforth refer to a CMF orbit as a twicemonotonic orbit.…”

Section: Outline Of the Methodsmentioning

confidence: 64%

Section: Numerical Integration: the Steepest-descent Methodsmentioning

confidence: 72%

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Two-degree-of-freedom Hamiltonian for the time-symmetric two-body problem of the relativistic action-at-a-distance electrodynamics

Hollander

Luca

2003

Phys. Rev. E

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The Rest-Frame Darwin Potential from the Lienard–Wiechert Solution in the Radiation Gauge

Crater¹,

Lusanna²

2001

Annals of Physics

184

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In the semiclassical approximation in which the electric charges of scalar particles are described by Grassmann variables (Q 2 i = 0, Q i Q j = 0), it is possible to re-express the Lienard-Wiechert potentials and electric fields in the radiation gauge as phase space functions, because the difference among retarded, advanced, and symmetric Green functions is of order Q 2 i . By working in the restframe instant form of dynamics, the elimination of the electromagnetic degrees of freedom by means of suitable second classs contraints leads to the identification of the Lienard-Wiechert reduced phase space containing only N charged particles with mutual action-at-a-distance vector and scalar potentials. A Darboux canonical basis of the reduced phase space is found. This allows one to re-express the potentials for arbitrary N as a unique effective scalar potential containing the Coulomb potential and the complete Darwin one, whose 1/c 2 component agrees with the known expression. The effective potential gives the classical analogue of all static and non-static effects of the one-photon exchange Feynman diagram of scalar electrodynamics. C

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Oscillation Criteria for Delay and Advanced Difference Equations with Variable Arguments

Stavroulakis

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Can the future influence the present?

Cited by 51 publications

References 13 publications

Two-degree-of-freedom Hamiltonian for the time-symmetric two-body problem of the relativistic action-at-a-distance electrodynamics

Two-degree-of-freedom Hamiltonian for the time-symmetric two-body problem of the relativistic action-at-a-distance electrodynamics

The Rest-Frame Darwin Potential from the Lienard–Wiechert Solution in the Radiation Gauge

Oscillation Criteria for Delay and Advanced Difference Equations with Variable Arguments

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