We obtain lagrangian for $k$-essence scalar field $\phi(r,t)$ with scalar curvature $k$ of Friedmann-Lemaitre-Robertson-Walker (FLRW) metric . Obtained lagrangian has two generalised co-ordinates $\phi$ and logarithm of scale factor ($q=\ln a$). Classical Hamiltonian $({ \mathcal H})$ is obtained in terms of two corresponding conjugate momentum $p_{ q}$ and $p_{ \phi}$. 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 $t\rightarrow\infty$, 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.
Categories: physics.gen-ph