On the matter Lagrangian of particles and fluids.

We consider a model where particles are described as localized concentrations of energy, with fixed rest mass and structure, which are not significantly affected by their self-induced gravitational field. We show that the volume average of the on-shell matter Lagrangian ${\mathcal L_m}$ describing such particles, in the proper frame, is equal to the volume average of the trace $T$ of the energy-momentum tensor in the same frame, independently of the particle's structure and constitution. Since both ${\mathcal L_m}$ and $T$ are scalars, and thus independent of the reference frame, this result is also applicable to collections of moving particles and, in particular, to those which can be described by a perfect fluid. Our results are expected to be particularly relevant in the case of modified theories of gravity with non-minimal coupling to matter where the matter Lagrangian appears explicitly in the equations of motion of the gravitational and matter fields, such as $f(R,{\mathcal L_m})$ and $f(R,T)$ gravity. In particular, they indicate that, in this context, $f(R,{\mathcal L_m})$ theories may be regarded as a subclass of $f(R,T)$ gravity.