Nonconservative Lie series: post-Newtonian binary dynamics at 2.5PN

We present a fully analytical solution to the dynamics of the non-spinning 2.5 post-Newtonian binary problem, accounting for both the long-term (secular) and short-term (oscillatory) temporal behavior, with no restriction on eccentricity. The radiative degrees of freedom are handled within the nonconservative Hamiltonian framework introduced in a companion paper. In this work, we apply the Lie series method to construct a resonant Birkhoff normal-form and the corresponding generator of the radiation-reaction dynamics. The secular piece reconstructs exactly the Peters-Mathews relations for semi-major axis and eccentricity. The oscillatory piece completes the dynamics and is well suited for gravitational wave templates. The procedure we present in this paper can be systematically employed to cast arbitrary nonconservative systems into extended Hamiltonian form so that the Lie method can be applied.

💡 Research Summary

The paper presents a fully analytical treatment of the dynamics of a non‑spinning compact binary system subject to the 2.5‑post‑Newtonian (PN) radiation‑reaction force, using a recently developed non‑conservative Hamiltonian framework together with Lie‑series perturbation theory. The authors begin by reviewing the limitations of traditional Lagrangian and Hamiltonian mechanics for dissipative processes and introduce the Galley‑type “doubled‑path” action principle, in which each physical degree of freedom q is duplicated into two copies (q↑, q↓). By forming average (q⁺) and difference (q⁻) variables, the non‑conservative effects are encoded in an extra term K(q↑, q↓, ẋ↑, ẋ↓) that couples the two paths. This construction yields an extended Hamiltonian A that lives on a symplectic manifold of doubled dimension and is antisymmetric under exchange of the two copies. Physical dynamics are recovered by imposing the physical‑limit condition q⁻→0, p⁻→0, which projects the doubled system back onto the original phase space.

The authors then specialize to the binary problem. The conservative part H₀ is taken as the 2‑PN ADM Hamiltonian, while the radiation‑reaction contribution appears at 2.5 PN order as a linear coupling R = q⁻·R_q(q⁺, p⁺)+p⁻·R_p(q⁺, p⁺). The small bookkeeping parameter ε tracks the 2.5 PN order. The total extended Hamiltonian reads A = H₀⁻ + ε A₁ + O(ε²), where H₀⁻ = H₀↑ − H₀↓ and A₁ contains the dissipative terms.

To obtain a normal‑form Hamiltonian, the authors employ a near‑identity canonical transformation generated by a Lie series T_g, with generator g = ε g₁ + ε² g₂ + …. In the purely conservative case the homological equation {g_ℓ, H₀}=P_ℓ can be solved straightforwardly. However, for the non‑conservative problem the doubled phase space introduces resonant angles θ⁻ = θ↑ − θ↓ because the two copies share the same frequencies when q↑≈q↓. These resonances prevent the complete removal of angle dependence from the normal form. The authors therefore formulate a “weak” homological equation that is satisfied up to terms of order O(−3) in the virtual variables (q⁻, p⁻). Since such terms vanish under the physical‑limit projection, they can be freely adjusted, allowing the construction of a resonant Birkhoff normal form.

The resulting normal‑form Hamiltonian separates into two pieces:

1. A non‑resonant part A*_NRES that depends only on the averaged actions K⁺ and reproduces exactly the secular evolution derived by Peters and Mathews (the classic ȧ(e) and ė(e) formulas). This confirms that the method captures the correct long‑term inspiral and circularisation.
2. A resonant part A*_RES that is linear in the resonant angles θ⁻ and drives a slow secular decay of the actions K⁺, embodying the dissipative back‑reaction.

Having obtained A*, the authors invert the Lie transformation to express the original variables (q⁺, p⁺) as explicit functions of time. The solution contains both the secular (orbit‑averaged) evolution and an oscillatory component arising from the higher‑order terms of the Lie series. The oscillatory piece encodes short‑timescale variations of the orbital elements that are absent in standard averaging methods but are crucial for constructing accurate gravitational‑wave templates, especially for eccentric binaries and for next‑generation detectors (e.g., LISA, Einstein Telescope).

The paper concludes by emphasizing the generality of the approach: any quasi‑integrable Hamiltonian system perturbed by weak dissipation can be cast into the extended Hamiltonian form, after which the same Lie‑series machinery applies. This opens the door to analytic treatments of a wide class of dissipative astrophysical problems, such as tidal friction, magnetic dipole radiation, or viscous fluid effects. Future work is suggested on extending the method to higher PN orders (3.5 PN, 4 PN), incorporating spin–orbit couplings, and benchmarking the analytical waveforms against numerical relativity simulations.