A Hamilton-like vector for the special-relativistic Coulomb problem

Muñoz, Gerardo; Pavić, Ivana

doi:10.1088/0143-0807/27/5/001

Cited by 14 publications

(10 citation statements)

References 15 publications

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“…This plane is always close to but, in general, not quite identical to the horizontal symmetry plane of the ring. In spherical ðr; θÞ coordinates the differential equation governing rðθÞ for a relativistic particle orbit in an inverse square law force field can be solved exactly [6]. These are like classical planetary orbits or the orbits in a hydrogen atom treated classically.…”

Section: Particle Tracking Paradigmsmentioning

confidence: 99%

“…Curved-planar electrodes produce the electric field shown in Figure 2 c. Solution of the equation of motion. Throughout much of this section formulas of Muñoz and Pavic [6] will be transcribed essentially unchanged, except for bringing symbols into consistency with conventional accelerator notation. The Muñoz/Pavic formulation, though equivalent to various other formalisms describing relativistic Coulomb orbits, is especially appropriate for our relativistic accelerator application.…”

Section: Relativistic Kinematics Inmentioning

confidence: 99%

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Symplectic orbit and spin tracking code for all-electric storage rings

Talman

2015

Phys. Rev. ST Accel. Beams

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Proposed methods for measuring the electric dipole moment (EDM) of the proton use an intense, polarized proton beam stored in an all-electric storage ring "trap." At the "magic" kinetic energy of 232.792 MeV, proton spins are "frozen," for example always parallel to the instantaneous particle momentum. Energy deviation from the magic value causes in-plane precession of the spin relative to the momentum. Any nonzero EDM value will cause out-of-plane precession-measuring this precession is the basis for the EDM determination. A proposed implementation of this measurement shows that a proton EDM value of 10 −29 e-cm or greater will produce a statistically significant, measurable precession after multiply repeated runs, assuming small beam depolarization during 1000 s runs, with high enough precision to test models of the early universe developed to account for the present day particle/antiparticle population imbalance. This paper describes an accelerator simulation code, ETEAPOT, a new component of the Unified Accelerator Libraries (UAL), to be used for long term tracking of particle orbits and spins in electric bend accelerators, in order to simulate EDM storage ring experiments. Though qualitatively much like magnetic rings, the nonconstant particle velocity in electric rings gives them significantly different properties, especially in weak focusing rings. Like the earlier code TEAPOT (for magnetic ring simulation) this code performs exact tracking in an idealized (approximate) lattice rather than the more conventional approach, which is approximate tracking in a more nearly exact lattice. The Bargmann-Michel-Telegdi (BMT) equation describing the evolution of spin vectors through idealized bend elements is also solved exactly-original to this paper. Furthermore the idealization permits the code to be exactly symplectic (with no artificial "symplectification"). Any residual spurious damping or antidamping is sufficiently small to permit reliable tracking for the long times, such as the 1000 s assumed in estimating the achievable EDM precision. This paper documents in detail the theoretical formulation implemented in ETEAPOT. An accompanying paper describes the practical application of the ETEAPOT code in the Universal Accelerator Libraries (UAL) environment to "resurrect," or reverse engineer, the "AGS-analog" all-electric ring built at Brookhaven National Laboratory in 1954. Of the (very few) all-electric rings ever commissioned, the AGS-analog ring is the only relativistic one and is the closest to what is needed for measuring proton (or, even more so, electron) EDM's. The companion paper also describes preliminary lattice studies for the planned proton EDM storage rings as well as testing the code for long time orbit and spin tracking.

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Section: Particle Tracking Paradigmsmentioning

confidence: 99%

Section: Relativistic Kinematics Inmentioning

confidence: 99%

Symplectic orbit and spin tracking code for all-electric storage rings

Talman

2015

Phys. Rev. ST Accel. Beams

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“…The only other instance known to the author with an attempt to identify and define a relativistic Hamilton vector is in a paper by Muñoz and Pavic [13] which also discusses extending Hamilton's method to relativistic Coulomb systems. Their approach, however, differs from the present paper in some major aspects :…”

Section: Alternative Definitions Of the Hamilton Vectormentioning

confidence: 99%

“…The success of the hodograph method for these systems prompts attempting using it also for relativistic charged particles in Coulomb fields. Although the relativistic spatial trajectories are much more complicated [12,13], it is indeed found that the same key feature -linearity of the velocity equation -persists into the relativistic regime, allowing relatively simple analytic discussion (Eqs. ( 13) & ( 15)).…”

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confidence: 99%

Hamilton symmetry in relativistic Coulomb systems

Ben-Ya'acov

2018

Preprint

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Hamilton's hodograph method geometrizes, in a simple and very elegant way, in velocity space, the full dynamics of classical particles in 1/r potentials. States of given energy and angular momentum are represented by circular hodographs whose radii depend only on the angular momentum, and hodographs differing only in the energy are related by uniform translations. This feature indicates the existence of an internal symmetry, named here after Hamilton. The hodograph method and the Hamilton symmetry are extended here for relativistic charged particles in a Coulomb field, on the relativistic velocity space which is a 3D hyperboloid H 3 embedded in a 3+1 pseudo-Euclidean space.The key for the simplicity and elegance of the velocity-space method is the linearity of the velocity equation, a unique feature of 1/r interactions for Newtonian and relativistic systems alike. Although with hodographs much more complicated than for Newtonian systems, the main features of the Hamilton symmetry persist in the full relativistic picture : (1) general hodographs may be represented as linearly displaced base energy-independent circles, (2) hodographs corresponding to same angular momentum but with different energies are connected via translations along geodesics of the velocity space. As an internal symmetry over and beyond central symmetry, the Hamilton symmetry is equivalent to the Laplace-Runge-Lenz symmetry and complements it.

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“…where the symbol | • | stands for the Euclidean norm of a two-dimensional vector, m is the mass of the particle, c is the speed of light and α is a constant (m, α > 0). Such an equation is well known in the physics community (see, for instance, [3,8,19] and the references therein); on the contrary, it has received much less attention by mathematicians working in the areas of Dynamical Systems and Nonlinear Analysis.…”

Section: Introductionmentioning

confidence: 99%