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Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach

We show how to systematically apply the Faddeev-Jackiw symplectic method to General Relativity (GR) and to GR extensions. This provides a new coherent frame for Hamiltonian analyses of gravitational theories. The emphasis is on the classical dynamics, uncovering the constraints, the gauge transformations and the number of degrees of freedom; but the method results are also relevant for canonical quantization approaches. We illustrate the method with three applications: GR and to two Brans-Dicke cases (the standard case $ω\not= - 3/2$ and the case with one less degree of freedom, $ω= - 3/2$). We clarify subtleties of the symplectic approach and comment on previous symplectic-based Hamiltonian analyses of extended theories of gravity, pointing out that the present approach is systematic, complete and robust.