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 $S_p=-mc\int ds+q\int A$, the canonical worldline one-form is $P=-m u^\flat+qA$, and a Killing vector satisfying $\mathcal L_XA=dχ_X$ 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.