  It has been shown by Remmen and Carroll that, for a model universe which
contains only a kinetically canonical scalar field minimally coupled to gravity
it is possible to choose 'special coordinates' to describe a two-dimensional
effective phase space. The special, non-canonical, coordinates are
$\phi$,$\dot{\phi}$ and the ability to describe an effective phase space with
these coordinates empowers the common usage of $\phi-\dot{\phi}$ as the space
to define inflationary initial conditions. This paper extends the result to the
full Horndeski action. The existence of a two-dimensional effective phase space
is shown for the general case. Subsets of the Horndeski action, relevant to
cosmology are considered as particular examples to highlight important aspects
of the procedure.
