= Charged-particle conserved quantity from a Killing symmetry
{title2=$mV_au^a-q\Phi$}
Suppose $V$ is a <Killing vector field>, $F$ is an <electromagnetic field tensor> satisfying $dF=0$ and $\mathcal L_VF=0$, and a particle of nonzero mass $m$ obeys the <Lorentz force> equation $\nabla_uu=(q/m)F^a{}_bu^b$. <Cartan's magic formula> gives $d(\iota_VF)=0$, so the <local potential of a closed differential one-form> proof gives $d\Phi=\iota_VF$ locally. The <Killing equation> eliminates $u^au^b\nabla_aV_b$, and the force term gives $d(V\cdot u)/d\tau=(q/m)d\Phi/d\tau$. Therefore $mV\cdot u-q\Phi$ is locally conserved. The sign assumes contraction in the first argument of $F$. A global potential additionally requires the closed one-form to have zero periods; invariance alone is insufficient without the homogeneous <Maxwell equations>.
Back to article page
