Gauge transformations of an Eisenhart-Duval lift
= Gauge transformations of an Eisenhart-Duval lift
{title2=$\mathbf A\mapsto\mathbf A+\nabla\chi,\quad u\mapsto u+2\chi$}
A time-independent <gauge transformation> $\mathbf A\mapsto\mathbf A+\nabla\chi$ in an <Eisenhart-Duval lift> is the coordinate change $u\mapsto u+2\chi$. It preserves the metric one-form $du-2\mathbf A\cdot d\mathbf x-Udt$ and adds only $-2d\chi/dt$ to the reduced Lagrangian.