The varied action (Route II)

Route II computes the variation of the frozen switched action independently of the coordinate problem, exposes every switch term, and records the classical Noether identity a later construction must respect. The gauge parameter is allowed to be an arbitrary compactly supported vector field in ${M_T}^{\circ }$, so that the switch terms are visible before the plateau condition is imposed.

For a compactly supported $\xi ,\zeta \in {C_c}^{\infty }({M_T}^{\circ };TM)$, let $F=F_{\kappa }$ be the genuine time-one flow of $\kappa \xi +{\kappa }^2\zeta$ and define the finite variation

The integrand is supported in the compact set where $F\ne \mathrm{id}$, so the integral is finite even though ${S_g}^{\mathrm{sc}}$ itself need not be.