  We present a simple but powerful Lagrangian method that can be used to study
the post-Newtonian evolution of a compact binary system with environment,
including a dark matter spike, around it, and obtain the resulting
gravitational wave emission. This formalism allows one to incorporate
post-Newtonian effects up to any desired known order, as well as any other
environmental effect around the binary, as long as their dissipation power or
force formulae are known. In particular, in this work, we employ this method to
study a black hole-black hole binary system of mass ratio $10^5$, by including
post-Newtonian effects of order 1PN and 2.5PN as well as the effect of
relativistic dynamical friction. We obtain the modified orbits and the
corresponding modified gravitational waveform. Finally, we contrast these
modifications against the LISA sensitivity curve in frequency space and show
that this observatory can detect the associated signals.
