Hamiltonian formalism in Friedmann cosmology and its quantization

We propose a Hamiltonian formalism for a generalized Friedmann-Robertson-Walker cosmology model in the presence of both a variable equation of state parameter $w(a)$ and a variable cosmological constant $\Lambda (a)$, where $a$ is the scale factor. This Hamiltonian system, containing 1 degree of freedom and without constraint, gives Friedmann equations as the equation of motion, which describes a mechanical system with a variable mass object moving in a potential field. After an appropriate transformation of the scale factor, this system can be further simplified to an object with constant mass moving in an effective potential field. In this framework, the $\Lambda$ cold dark matter model as the current standard model of cosmology corresponds to a harmonic oscillator. We further generalize this formalism to take into account the bulk viscosity and other cases. The Hamiltonian can be quantized straightforwardly, but this is different from the approach of the Wheeler-DeWitt equation in quantum cosmology.