Dynamical similarity in field theories

In previous work I have shown that Herglotz actions reproduce the dynamics of classical mechanical theories which exhibit dynamical similarities. Recent work has shown how to extend field theories in both the Lagrangian and de Donder-Weyl formalism to contact geometry (Gaset et al 2020 Ann. Phys., NY414 168092; 2021 Rep. Math. Phys.87 347–68; 2022 arXiv:2211.17058). In this article I show how dynamical similarity applies in field theory. This is applied in both the Lagrangian and Hamiltonian frameworks, producing the contact equivalents. The result can be applied to general relativity where I demonstrate how to construct a complete description of the dynamics, equivalent to those derived from the Einstein–Hilbert action, without reference to the conformal factor.