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A novel routine to investigate the scalar fields in a cosmological context isdiscussed in the framework of the Hamiltonian formalism. Starting from theEinstein-Hilbert action coupled to a Lagrangian density that contains twocomponents - one corresponding to a scalar field Lagrangian,${\mathcal{L}}_{\phi}$, and another that depends on the scale parameter,${\mathcal{L}}_{a}$ - one can identify a generalized Hamiltonian density fromwhich first-order dynamical equations can be obtained. This set up correspondsto the dynamics of Friedmann-Robertson-Walker models in the presence ofhomogeneous fields embedded into a generalized cosmological background fluid ina system that evolves all together isentropically. Once the generalizedHamiltonian density is properly defined, the constraints on thegravity-matter-field system are straightforwardly obtained through thefirst-order Hamilton equations. The procedure is illustrated for three examplesof cosmological interest for studies of the dark sector: real scalar fields,tachyonic fields and generalized Born-Infeld tachyonic fields. The inclusion ofsome isentropic fluid component into the Friedmann equation allows foridentifying an exact correspondence between the dark sector underlying scalarfield and an ordinary real scalar field dynamics. As a final issue, theHamiltonian formulation is used to set the first-order dynamical equationsthrough which one obtains the exact analytical description of the cosmologicalevolution of a generalized Chaplygin gas (GCG) with dustlike matter, radiationor curvature contributions. Model stability in terms of the square of the soundvelocity, $c_{s}^{2}$, cosmic acceleration, $q$, and conditions for inflationare discussed.

The Hamiltonian formalism for scalar fields coupled to gravity in a cosmological background