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 ${\mathcal{L}}_{[p]}=T_{[p]}\mathrm{/}(p+1)$, with $T_{[p]}$ 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 ${\mathcal{L}}_{[0]}=T_{[0]}$ 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 $f(R,{\mathcal{L}}_{\mathrm{m}})$ 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-Lemaître-Robertson-Walker spacetimes.