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

We obtain lagrangian for $k$-essence scalar field $\varphi (r,t)$ with scalar curvature $k$ of Friedmann-Lemaitre-Robertson-Walker (FLRW) metric . Obtained lagrangian has two generalised co-ordinates $\varphi$ and logarithm of scale factor ($q=\ln a$). Classical Hamiltonian $(\mathcal{H})$ is obtained in terms of two corresponding conjugate momentum $p_q$ and $p_{\varphi }$. Solving Hamilton’s equation of motion , we obtain classical solution for scale factor $a(t)$, energy density $\rho$, equation of state parameter $\omega$ and deceleration parameter $q_0$ . At late time as $t\to \mathrm{\infty }$, we have an exponential growth of scale factor with time, energy density $\rho$ becomes constant, which we can identify as dark energy density, equation of state parameter becomes $\omega \approx -1$ 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.