  (Abridged): The standard adiabatic approximation to phasing of gravitational
waves from inspiralling compact binaries uses the post-Newtonian expansions of
the binding energy and gravitational wave flux both truncated at the same
relative post-Newtonian order. Motivated by the eventual need to go beyond the
adiabatic approximation we must view the problem as the dynamics of the binary
under conservative post-Newtonian forces and gravitational radiation damping.
  From the viewpoint of the dynamics of the binary, the standard approximation
at leading order is equivalent to retaining the 0PN and 2.5PN terms in the
acceleration and neglecting the intervening 1PN and 2PN terms. A complete
mathematically consistent treatment of the acceleration at leading order should
include all PN terms up to 2.5PN without any gaps. These define the 'standard'
and 'complete' non-adiabatic approximants respectively. We propose a new and
simple complete adiabatic approximant constructed from the energy and flux
functions. At the leading order it uses the 2PN energy function rather than the
0PN one in the standard approximation so that in spirit it corresponds to the
dynamics where there are no missing post-Newtonian terms in the acceleration.
We compare the overlaps of the standard and complete adiabatic approximants
with the exact waveforms for a test-particle orbiting a Schwarzschild black
hole. The complete adiabatic approximants lead to a remarkable improvement in
the effectualness at lower PN (< 3PN) orders. However, standard adiabatic
approximants of order $\geq$ 3PN are nearly as good as the complete adiabatic
approximants for the construction of effectual templates. Standard and complete
approximants beyond the adiabatic approximation are next studied using the
Lagrangian models of Buonanno, Chen and Vallisneri.
