Can the Lagrangian density of a field be derived from the stress-energy tensor?

I have been learning some field theory and learning about Lagrangian and Hamiltonian density. In classical mechanics, the Hamiltonian is the energy of the system in terms of position and momentum. I understand that similarly, the Hamiltonian density is the energy density in terms of the field and the conjugate momentum field. In that way, the Hamiltonian density is $T_{00}$, where $T$ is the stress-energy tensor.

Since the Lagrangian density is relativistically invariant, is there an invariant relation between Lagrangian density and the stress-energy tensor?