Logarithm of the scale factor as a generalised coordinate in a lagrangian for dark matter and dark energy

A lagrangian for the $k-$ essence field is set up with canonical kinetic terms and incorporating the scaling relation of [1]. There are two degrees of freedom, {\it viz.},$q(t)=lna(t)$ ($a(t)$ is the scale factor) and the scalar field $\varphi$, and an interaction term involving $\varphi$ and $q(t)$.The Euler-Lagrange equations are solved for $q$ and $\varphi$. Using these solutions quantities of cosmological interest are determined. The energy density $\rho$ has a constant component which we identify as dark energy and a component behaving as $a^{-3}$ which we call dark matter. The pressure $p$ is {\it negative} for time $t\to \mathrm{\infty }$ and the sound velocity $c_{s}^{2}={\partial p\over\partial\rho} << 1$. When dark energy dominates, the deceleration parameter $Q\to -1$ while in the matter dominated era $Q\sim \frac{1}{2}$. The equation of state parameter $w=\frac{p}{\rho }$ is shown to be consistent with $w=\frac{p}{\rho }\sim -1$ for dark energy domination and during the matter dominated era we have $w\sim 0$. Bounds for the parameters of the theory are estimated from observational data. Keywords: k-essence models, dark matter, dark energy PACS No: 98.80.-k