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
nonminimal 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.
