k-essence classical Hamiltonian approach for an accelerated expansion of the Universe with $ω\approx-1$

We obtain lagrangian for $k$-essence scalar field $ϕ(r,t)$ with scalar curvature $k$ of Friedmann-Lemaitre-Robertson-Walker (FLRW) metric . Obtained lagrangian has two generalised co-ordinates $ϕ$ and logarithm of scale factor ($q=\ln a$). Classical Hamiltonian $({\mathcal H})$ is obtained in terms of two corresponding conjugate momentum $p_{q}$ and $p_ϕ$. Solving Hamilton's equation of motion , we obtain classical solution for scale factor $a(t)$, energy density $ρ$, equation of state parameter $ω$ and deceleration parameter $q_{0}$ . At late time as $t\rightarrow\infty$, we have an exponential growth of scale factor with time, energy density $ρ$ becomes constant, which we can identify as dark energy density, equation of state parameter becomes $ω\approx -1$ and deceleration parameter becomes $q_{0}\approx -1$. All this results indicates an accelerated expansion of universe driven by negative pressure known as dark energy.

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