From gauge-covariant Noether invariants to electromagnetic velocity-dependent mechanics
Abstract
For a charged particle, a spacetime symmetry need preserve the electromagnetic potential only up to a gauge transformation. Starting from the standard Noether--Bessel-Hagen construction, we make explicit the reduction from the relativistic gauge-invariant worldline charge to the generalized constants of nonrelativistic velocity-dependent mechanics, retaining the quasi-invariance boundary term throughout. With the convention Sp=−mc∫ ds+q∫ A, the canonical worldline one-form is P=−m u♭+qA, and a Killing vector satisfying ℒXA=dχX generates CX=P(X)−qχX. Its low-velocity limit is the Noether--Bessel-Hagen charge of the coordinate-time Lagrangian, from which generalized linear momentum, angular momentum and energy follow generator by generator. A uniform magnetic field gives an elementary gauge-dependent representative but gauge-invariant translational charge. The Darwin interaction then supplies the conservative O(c−2) many-particle realization after field elimination, while a closed particle--field system is governed by the total stress-energy Killing charge. The result is an explicit reduction map linking three established descriptions without identifying their distinct conserved quantities.
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BibTeX
@misc{castro2026from,
title = {{From gauge-covariant Noether invariants to electromagnetic velocity-dependent mechanics}},
author = {A. S. de Castro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38128},
eprint = {2610.38128},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38128},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From gauge-covariant Noether invariants to electromagnetic velocity-dependent mechanics
AU - A. S. de Castro
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38128
UR - https://arxiv.org/abs/2610.38128
AB - For a charged particle, a spacetime symmetry need preserve the electromagnetic potential only up to a gauge transformation. Starting from the standard Noether--Bessel-Hagen construction, we make explicit the reduction from the relativistic gauge-invariant worldline charge to the generalized constants of nonrelativistic velocity-dependent mechanics, retaining the quasi-invariance boundary term throughout. With the convention S_p=−mc∫ ds+q∫ A, the canonical worldline one-form is P=−m u^♭+qA, and a Killing vector satisfying ℒ_XA=dχ_X generates C_X=P(X)−qχ_X. Its low-velocity limit is the Noether--Bessel-Hagen charge of the coordinate-time Lagrangian, from which generalized linear momentum, angular momentum and energy follow generator by generator. A uniform magnetic field gives an elementary gauge-dependent representative but gauge-invariant translational charge. The Darwin interaction then supplies the conservative O(c^−2) many-particle realization after field elimination, while a closed particle--field system is governed by the total stress-energy Killing charge. The result is an explicit reduction map linking three established descriptions without identifying their distinct conserved quantities.
ER -