uncommon questions - Lagrange function of zero

In classical mechanics it is difficult to imagine that the Lagrange function \(\mathcal{L}\) of a system would be zero in a non-trivial way; one example could be the motion of a particle at constant velocity, because any constant can be made to vanish by using the invariance of the Euler-Lagrange-equation under the transformation \(\mathcal{L}\rightarrow a\mathcal{L}+b\), or by Galilei-boosting into the rest frame of the particle.

Can you find examples of naturally vanishing Lagrange-functions or Lagrange-densities in relativistic mechanics and relativistic field theories and to what physical system would they correspond?