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 $p=0$ this reduces to the standard $\mathcal{L}_{\rm [0]}=T_{\rm [0]}$ relation, which determines the on-shell Lagrangian of point particles and their fluids, more generally it depends explicitly on the $p$-brane dimensionality. We explore the implications of this Lagrangian identity for the dynamics of non-self-intersecting cosmic string loops in a homogeneous and isotropic universe within $f(R,\mathcal{L}_{\rm m})$ gravity, showing that, unlike in general relativity, the proper mass of a cosmic string loop may evolve over cosmological timescales regardless of its small size or tension. Finally, we extend the analysis to the more general case of closed $p$-branes in $(N+1)$-dimensional Friedmann-Lema\^itre-Robertson-Walker spacetimes.