Lagrangian Identity and Mass Evolution of Particle-like Objects in Nonminimally Coupled Gravity

We show that the Lagrangian of a Nambu-Goto $p$-brane satisfies the identity $\mathcal{L}_{\rm [\it p \rm]}=T_{\rm [\it p \rm]}/(p+1)$, with $T_{\rm [\it p \rm]}$ denoting the trace of the corresponding energy-momentum tensor, independently of the properties of the gravitational field. While for $p=