  We introduce a formulation for spinning gravitating objects in the effective
field theory in the post-Newtonian scheme in the context of the binary inspiral
problem. We aim at an effective action, where all field modes below the orbital
scale are integrated out. We spell out the relevant degrees of freedom, in
particular the rotational ones, and the associated symmetries. Building on
these symmetries, we introduce the minimal coupling part of the point particle
action in terms of gauge rotational variables, and construct the spin-induced
nonminimal couplings, where we obtain the leading order couplings to all orders
in spin. We specify the gauge for the rotational variables, where the
unphysical degrees of freedom are eliminated already from the Feynman rules,
and all the orbital field modes are integrated out. The equations of motion of
the spin can be directly obtained via a proper variation of the action, and
Hamiltonians may be straightforwardly derived. We implement this effective
field theory for spin to derive all spin dependent potentials up to
next-to-leading order to quadratic level in spin, namely up to the third
post-Newtonian order for rapidly rotating compact objects. In particular, the
proper next-to-leading order spin-squared potential and Hamiltonian for generic
compact objects are also derived. For the implementations we use the
nonrelativistic gravitational field decomposition, which is found here to
eliminate higher-loop Feynman diagrams also in spin dependent sectors, and
facilitates derivations. This formulation for spin is thus ideal for treatment
of higher order spin dependent sectors.
