Dynamics of Relativistic Binaries in Structured and Stochastic Environments: A Lagrange-Fourier-Hansen Framework

We develop a general framework to characterize non-vacuum perturbations to relativistic binaries in the gravitational-wave (GW) driven regime, for use in GW parameter estimation studies. The effect of smooth, structured and stochastic perturbations to the binary's motion is reduced to a resonant spectral projection defined on a rolling averaging window, with weights given by Hansen coefficients. This is combined with practical criteria for identifying and evaluating the corresponding dynamical response to perturbations, starting from either analytical models or numerical simulations of binaries in environments. The result is a set of coupled ODEs for the orbital elements that capture epi-cyclic, apsidal and nodal resonances, consistently incorporate feedback from radiation reaction and can be solved efficiently on a coarse time grid. We demonstrate the practical application of the framework in two representative astrophysical scenarios: a compact binary in a variable tidal field and an extreme-mass-ratio inspiral in an accretion disk. We propose the Lagrange-Fourier-Hansen framework as a unified tool for modeling environmental effects in GW templates for eccentric and precessing binary sources, and particularly for bridging the gap between phenomenological prescriptions and realistic models of binaries in environments.