**Publication date:** Nov 2021

**Abstract:**

In this letter a new Lagrangian variational principle is proved to hold for the Einstein field equations, in which the independent variational tensor field is identified with the Ricci curvature tensor R^{μ ν} rather than the metric tensor g_{μ ν}. The corresponding Lagrangian function, denoted as L_{R}, is realized by a polynomial expression of the Ricci 4-scalar R ≡g_{μ ν}R^{μ ν} and of the quadratic curvature 4-scalar ρ ≡R^{μ ν}R_{μ ν} . The Lagrangian variational principle applies both to vacuum and non-vacuum cases and for its validity it demands a non-vanishing, and actually also positive, cosmological constant Λ >0 . Then, by implementing the deDonder-Weyl formalism, the physical conditions for the existence of a manifestly-covariant Hamiltonian structure associated with such a Lagrangian formulation are investigated. As a consequence, it is proved that the Ricci tensor can obey a Hamiltonian dynamics which is consistent with the solutions predicted by the Einstein field equations.

**Authors:**

Cremaschini, Claudio; Kovář, Jiří; Stuchlík, Zdeněk; Tessarotto, Massimo;