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We obtain lagrangian for $k$-essence scalar field $\phi(r,t)$ with scalarcurvature $k$ of Friedmann-Lemaitre-Robertson-Walker (FLRW) metric . Obtainedlagrangian has two generalised co-ordinates $\phi$ and logarithm of scalefactor ($q=\ln a$). Classical Hamiltonian $({\mathcal H})$ is obtained in termsof two corresponding conjugate momentum $p_{q}$ and $p_{\phi}$. SolvingHamilton's equation of motion , we obtain classical solution for scale factor$a(t)$, energy density $\rho$, equation of state parameter $\omega$ anddeceleration parameter $q_{0}$ . At late time as $t\rightarrow\infty$, we havean exponential growth of scale factor with time, energy density $\rho$ becomesconstant, which we can identify as dark energy density, equation of stateparameter becomes $\omega\approx -1$ and deceleration parameter becomes $q_{0}\approx -1$. All this results indicates an accelerated expansion ofuniverse driven by negative pressure known as dark energy.

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