Two-Timescale Averaging

Two-timescale averaging replaces a weakly forced quasiperiodic orbit by slowly evolving actions and uniformly valid phases. A near-identity transformation removes nonresonant oscillatory forcing order by order; the zero Fourier mode survives as secular drift. The construction is controlled only when its frequency denominators stay away from zero and the slow evolution remains separated from the fast periods.

Helpful background. The Einstein Equation fixes the dynamics, Weak Metric Perturbations and Gauge Freedom supplies perturbative variables, Presymplectic Potentials and Currents supplies action variables, Partial Differential Equations and Green Functions supplies retarded forcing, Gravitational Radiation and Waveforms supplies flux checks, and Relativistic Two-Body Dynamics supplies the orbital regimes.

Fast phases and slow actions

Let $\epsilon\ll1$ measure radiation reaction. In action–angle variables,

$$\dot q^\alpha =\Omega^\alpha(J) +\epsilon g^\alpha(q,J)+O(\epsilon^2),$$ $$\dot J_A =\epsilon G_A(q,J) +\epsilon^2G_A^{(2)}(q,J)+O(\epsilon^3).$$

Expand a periodic forcing on the angle torus:

$$G_A(q,J) =\sum_{\mathbf k\in\mathbb Z^d} G_{A,\mathbf k}(J)e^{i\mathbf k\cdot\mathbf q}.$$

The torus average is the zero mode,

$$\langle G_A\rangle =G_{A,\mathbf0} =\frac{1}{(2\pi)^d} \int d^dq\,G_A(q,J).$$

For nonresonant motion, long-time and torus averages agree. At a resonance $\mathbf k\cdot\boldsymbol\Omega=0$, a trajectory samples only a lower dimensional subset and they differ.

The integration is over $0\le q^\alpha\lt2\pi$ for every angle. These angles must advance uniformly on each unforced torus with respect to the chosen time variable; averaging uniformly over a geometrical anomaly or mixing Mino-time and coordinate-time averages uses a different measure.

Near-identity transformation

Seek transformed variables

$$\widetilde J_A =J_A+\epsilon Y_A(q,J)+O(\epsilon^2), \qquad \widetilde q^\alpha =q^\alpha+\epsilon X^\alpha(q,J)+O(\epsilon^2).$$

For each nonzero Fourier mode, choosing

$$Y_{A,\mathbf k} =-\frac{G_{A,\mathbf k}} {i\mathbf k\cdot\boldsymbol\Omega}$$

removes the oscillatory $O(\epsilon)$ forcing. The averaged equation becomes

$$\dot{\widetilde J}_A =\epsilon G_{A,\mathbf0}(\widetilde J) +O(\epsilon^2).$$

The denominator shows both the method and its boundary. If a low-order combination of frequencies approaches zero, $Y_A$ is no longer near identity. A resonant inner expansion must retain that harmonic.

For infinitely many Fourier modes, nonzero denominators individually are not sufficient: their inverses must be controlled relative to the decay of the Fourier coefficients. A smooth finite-mode truncation with a positive minimum denominator is the simplest bounded setting for the calculation above. Stronger infinite-series statements require regularity and small-denominator estimates.

At the next order, the average includes $G_A^{(2)}$ and quadratic combinations of the first-order oscillatory terms and derivatives of $\Omega$. These terms contribute to first-post-adiabatic phase even when their instantaneous force is one order smaller.

The general two-timescale construction for relativistic inspirals is developed in Hinderer and Flanagan 2008, §§III–VI.

Solvability and uniform phase

A naive expansion at fixed time produces secular terms such as $\epsilon t$ and fails when $t=O(\epsilon^{-1})$. Introduce slow time

$$\widetilde t=\epsilon t$$

and allow the leading actions and phases to depend on it. At each order, the zero-mode equation is the solvability condition that prevents an unbounded periodic correction. The phase has the structure

$$\Phi(t) =\epsilon^{-1}\Phi_{-1}(\widetilde t) +\Phi_0(\widetilde t) +O(\epsilon).$$

Leading averaged dissipation determines $\Phi_{-1}$. Conservative and oscillatory first-order effects and averaged second-order dissipation enter $\Phi_0$. Dropping them can leave an order-unity phase error over one radiation-reaction time.

Direct-integration benchmark

First solve an example exactly. With constant $\omega\ne0$, let

$$\dot q=\omega, \qquad \dot J=\epsilon(a+b\cos q), \qquad (q,J)_{t=0}=(q_0,J_0).$$

Direct integration gives

$$J(t)=J_0+\epsilon at +\frac{\epsilon b}{\omega} \bigl[\sin(q_0+\omega t)-\sin q_0\bigr].$$

The averaging transformation is $\widetilde J=J-\epsilon b\sin q/\omega$, so $\dot{\widetilde J}=\epsilon a$ exactly. The matching initial value is $\widetilde J(0)=J_0-\epsilon b\sin q_0/\omega$, not $J_0$. Reconstructing $J=\widetilde J+\epsilon b\sin q/\omega$ recovers the full solution for arbitrarily long times. This example shows separately the secular drift, the bounded fast oscillation, and the initial-data shift. In a system where $\Omega$ depends on $J$, an uncorrected $O(\epsilon)$ initial action offset also produces an $O(1)$ phase error over an inspiral.

Use a nonresonant forced oscillator with known Fourier coefficients. Starting from matched physical initial data:

1. integrate the full equations and the averaged equations to $t=C/\epsilon$;
2. reconstruct physical variables with the near-identity transformation;
3. compare actions and unwrapped phases;
4. repeat for decreasing $\epsilon$.

With a first-order averaged model, reconstructed action error should follow the predicted uniform order away from resonance, while omitting reconstruction leaves an $O(\epsilon)$ oscillation. A phase comparison must not re-fit a time-dependent alignment, which would hide secular error.

The method fails near small denominators, separatrices with divergent periods, overlapping resonances, and abrupt forcing. It also requires enough regularity of the forcing on the torus. Flux balance may determine the averaged dissipative actions, but it does not supply the oscillatory transformation needed to reconstruct local motion.

Independent checks

The most discriminating check compares three objects, not two: direct integration, the averaged variables, and the reconstructed physical variables. The raw averaged actions should miss the bounded fast oscillation, while near-identity reconstruction restores it with the predicted residual order. This separates an averaging error from a reconstruction error.

One should also compute every small denominator along the trajectory. If the minimum $\lvert\mathbf k\cdot\boldsymbol\Omega\rvert$ shrinks in proportion to $\sqrt\epsilon$ for a retained low-order harmonic, the nonresonant error bound is no longer uniform even if the numerical trajectories happen to look smooth. Finally, set the forcing to its torus average by hand: the oscillatory generators must vanish, and direct and averaged evolution must then agree without reconstruction through the retained order.

Exercises

Derive the coefficient $Y_{A,\mathbf k}$ that removes a nonzero Fourier mode.

Solution

To leading order,

$$\frac{dY_A}{dt} =\Omega^\alpha\partial_{q^\alpha}Y_A.$$

For one mode this is $i(\mathbf k\cdot\boldsymbol\Omega)Y_{A,\mathbf k}e^{i\mathbf k\cdot q}$. Canceling $G_{A,\mathbf k}e^{i\mathbf k\cdot q}$ gives the displayed negative ratio. It exists only when the frequency combination is nonzero.

Why does an $O(\epsilon^2)$ drift affect phase at order unity?

Solution

Across a time $O(\epsilon^{-1})$, the drift changes an action and hence a frequency by $O(\epsilon)$. Integrating that frequency correction for another $O(\epsilon^{-1})$ factor of time yields an $O(1)$ phase. Secular integration promotes the apparently smaller force term.

References

• Hinderer, Tanja, and Éanna É. Flanagan. “Two Timescale Analysis of Extreme Mass Ratio Inspirals in Kerr. I. Orbital Motion.” Physical Review D 78 (2008): 064028. DOI.