Leo C. Stein

Leo C. Stein

Physics Professor, U of MS. Black holes, gravitational waves, general relativity and beyond.

Notes: Near-identity transformations to split fast and slow motion

The name near-identity transformation (NIT) is just shorthand for a specific application of perturbation theory—one which is particularly useful in dynamical systems that have slow and fast timescales. This is not all they can be used for (see e.g. Fumagalli+1 where a NIT is used to remove gauge dependence). What I write below is covered in some standard references but it’s easy enough to rederive it, so I’m writing it here so I can easily find my derivation.

The nicest possible setting where we can apply a NIT is close to an integrable Hamiltonian system. Say we start with a system we can put into action-angle variables,

\begin{align} \label{eq:AA-EOM} \dot{J}_a &= 0 , & \dot{\phi}^a &= \omega^a(\bs{J}) , \end{align}

where the ’s are all -periodic, so the phase space is foliated by N-tori. This system already has a slow-fast split; in fact the action variables don’t evolve at all, which is the slowest possible! But, now we add a perturbation that breaks this structure and add forcing terms to the right hand sides,

\begin{align} \label{eq:pertEOMJ} \dot{J}_a &= \epsilon F_a(\bs{J}, \bs{\phi}) , \\ \label{eq:pertEOMphi} \dot{\phi}^a &= \omega^a(\bs{J}) + \epsilon f^a(\bs{J}, \bs{\phi}) . \end{align}

This doesn’t need to be a Hamiltonian system. (Here I will only do first order. This can be developed to arbitrary order; see e.g. Lynch+2 for second order in the context of extreme mass-ratio inspirals, but beware of differences in notation).

The trouble with Eqs. \eqref{eq:pertEOMJ} and \eqref{eq:pertEOMphi} is that now both the ’s and the ’s can vary rapidly on the short timescale, while there is also (usually) a slow secular drift. A classic way to get a simpler dynamical system is to simply average the right hand sides. This is covered in most standard textbooks like Goldstein or Jose and Saletan. But, it turns out we can do much better. First, we give ourselves an infinite amount of additional freedom by doubling the number of functions we’re using,3

\begin{align} \label{eq:bar-tilde-split} \bs{J}(t) &= \bar{\bs{J}}(t) + \epsilon \tilde{\bs{J}}(t) , & \bs{\phi}(t) &= \bar{\bs{\phi}}(t) + \epsilon \tilde{\bs{\phi}}(t) . \end{align}

Well, obviously we’re going to need twice as many equations. But we basically get to pick whatever extra equations we want to close the system, which is why NITs are so powerful: you can use them to enforce a slow/fast split; or as in Fumagalli+,1 pull out gauge dependence; or maybe something else!

Anyway, a convenient choice for the variables is that they just evolve on a slow timescale, which is longer than the fast time given by . Let’s decompose the ’s and ’s as Fourier series on their N-tori,

\begin{align} F_a(\bs{J}, \bs{\phi}) &= \sum_{\vec{n}\in \mathbb{Z}^N} F_{a,\vec{n}}(\bs{J}) \exp(i \vec{n}\cdot\vec{\phi} ) , \\ f^a(\bs{J}, \bs{\phi}) &= \sum_{\vec{n}\in \mathbb{Z}^N} f^a_{\vec{n}}(\bs{J}) \exp(i \vec{n}\cdot\vec{\phi} ) . \end{align}

The mode with is just the torus-average, which is -independent, so we can write the split

\begin{align} f^a(\bs{J}, \bs{\phi}) &= \avg{f^a}(\bs{J}) + \sum_{\vec{n}\neq \bs{0}} f^a_{\vec{n}}(\bs{J}) e^{i \vec{n}\cdot\vec{\phi}} , \end{align}

where the average is simply

\begin{align} \avg{f^a}(\bs{J}) \equiv \int \frac{d^N\phi}{(2\pi)^N} f^a(\bs{J},\bs{\phi}) \end{align}

and similarly for the ’s.

Now we plug in this Fourier decomposition and the bar/tilde split from Eq. \eqref{eq:bar-tilde-split} into the equations of motion,

\begin{align} \label{eq:J-bar-tilde-EOM} \dot{\bar{J}}_a + \epsilon \dot{\tilde{J}}_a &= \epsilon \avg{F_a}(\bs{J}) + \epsilon \sum_{\vec{n}\neq \bs{0}} F_{a,\vec{n}}(\bs{J}) e^{i \vec{n}\cdot\vec{\phi}} , \\ \label{eq:phi-bar-tilde-EOM} \dot{\bar{\phi}}^a + \epsilon \dot{\tilde{\phi}}^a &= \omega^a(\bar{\bs{J}}+\epsilon \tilde{\bs{J}}) + \epsilon \avg{f^a}(\bs{J}) + \epsilon \sum_{\vec{n}\neq \bs{0}} f^a_{\vec{n}}(\bs{J}) e^{i \vec{n}\cdot\vec{\phi}} . \end{align}

Note that I wrote . We can expand

\begin{align} \omega^a(\bar{\bs{J}} + \epsilon\tilde{\bs{J}}) = \omega^a(\bar{\bs{J}}) + \epsilon \tilde{J}_b \frac{\pd \omega^a}{\pd J_b}(\bar{\bs{J}}) . \end{align}

Everywhere else on the RHSs of Eqs. \eqref{eq:J-bar-tilde-EOM} and \eqref{eq:phi-bar-tilde-EOM}, the difference between and is higher order than we need to track, so we can safely replace with when convenient. Similarly, we can replace with when convenient.

The first thing to notice is that if we choose the equation for to be the averaged one,

\begin{align} \label{eq:bar-J-dot} \dot{\bar{J}}_a = \epsilon \ \avg{F_a}(\bar{\bs{J}}) , \end{align}

then the form an autonomous system that evolves only on the slow time—there is no short-time oscillatory dependence here. These are the “slow” variables with secular effects. Similarly, it would be convenient for the equation of motion for the variables to only depend on on the right hand side, similar to the action-angle equations of motion \eqref{eq:AA-EOM}. We are free to choose

\begin{align} \label{eq:bar-phi-dot} \dot{\bar{\phi}}^a = \omega^a(\bar{\bs{J}}) + \epsilon \ \avg{f^a}(\bar{\bs{J}}) . \end{align}

So the phases evolve with slowly-varying frequencies. The system of Eqs. \eqref{eq:bar-J-dot} and \eqref{eq:bar-phi-dot} can be easily evolved over long timescales, taking rather large timesteps.

Subtracting from the original equations of motion \eqref{eq:J-bar-tilde-EOM} and \eqref{eq:phi-bar-tilde-EOM}, the tilde variables must satisfy

\begin{align} \label{eq:tilde-J-dot} \dot{\tilde{J}}_a &= \sum_{\vec{n}\neq \bs{0}} F_{a,\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \\ \label{eq:tilde-phi-dot} \dot{\tilde{\phi}}^a &= \tilde{J}_b \frac{\pd \omega^a}{\pd \bar{J}_b}(\bs{\bar{J}}) + \sum_{\vec{n}\neq \bs{0}} f^{a}_{\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \end{align}

where we have replaced unbarred with barred variables on the right hand sides, because everything here is already . Observe that the RHS of \eqref{eq:tilde-J-dot} has zero mean, so except for a constant of integration—that we choose to vanish—we know that are purely oscillatory, with no secular effects. Let’s perform a Fourier decomposition of and compute its time derivative,

\begin{align} \tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \\ \label{eq:tilde-J-Fourier-dot} \frac{d}{dt}\tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \left[ \frac{\pd \tilde{J}_{a,\vec{n}}}{\pd \bar{J}_b} \dot{\bar{J}}_b + \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) (i \vec{n}\cdot \dot{\bar{\bs{\phi}}}) \right]e^{i \vec{n}\cdot\bar{\phi}} , \\ \label{eq:tilde-J-Fourier-dot-plug-in} \frac{d}{dt}\tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) (i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})) e^{i \vec{n}\cdot\bar{\phi}} + \mathcal{O}(\epsilon) . \end{align}

In going from Eq. \eqref{eq:tilde-J-Fourier-dot} to \eqref{eq:tilde-J-Fourier-dot-plug-in}, we plugged in the equations of motion for and , keeping only the piece. Now we simply match coefficients in this Fourier expansion and the RHS of Eq. \eqref{eq:tilde-J-dot} to find

\begin{align} \label{eq:tilde-J-sol} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) = \frac{F_{a,\vec{n}}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} . \end{align}

Notice that this will fail near resonances, where might no longer be small relative to ! But anyway, far from (important) resonances, if you know the Fourier decomposition of the forcing functions on your tori, then you know the solution for (once you plug in a (possibly numerical) solution for ).

The same approach works for , except there is one more term in Eq. \eqref{eq:tilde-phi-dot}. In the term , it is important that depends only on , so that its Fourier expansion is purely “DC”. This means we don’t have to re-expand a product of Fourier expansions. Fourier-expanding , taking a time derivative, again using the time derivatives and , and equating with the RHS of Eq. \eqref{eq:tilde-phi-dot}, we eventually find

\begin{align} \tilde{\phi}^a_{\vec{n}}(\bar{\bs{J}}) (i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})) &= \tilde{J}_{b,\vec{n}}(\bar{\bs{J}}) \frac{\pd \omega^a}{\pd \bar{J}_b}(\bar{\bs{J}}) + f^a_\vec{n}(\bar{\bs{J}}) , \\ \tilde{\phi}^a_{\vec{n}}(\bar{\bs{J}}) &= \frac{F_{b,\vec{n}}(\bar{\bs{J}})}{(i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}}))^2} \frac{\pd \omega^a}{\pd \bar{J}_b}(\bar{\bs{J}}) + \frac{f^a_\vec{n}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} , \end{align}

where in going to the second line we plugged in the solution for from Eq. \eqref{eq:tilde-J-sol}. Here of course we still have the problem of small denominators near resonances; but other than that, the solution is just given in terms of Fourier coefficients, frequencies, and (background) gradients of the frequencies, all evaluated upon the slow solution for .

Summary

Summarizing, the full solution—with both secular drifts and oscillations on short timescales—is reconstructed from the sums

\begin{align} \bs{J}(t) &= \bar{\bs{J}}(t) + \epsilon \tilde{\bs{J}}(t) , & \bs{\phi}(t) &= \bar{\bs{\phi}}(t) + \epsilon \tilde{\bs{\phi}}(t) , \end{align}

where the barred (slow) variables solve the system

\begin{align} \dot{\bar{J}}_a &= \epsilon \ \avg{F_a}(\bar{\bs{J}}) , \\ \dot{\bar{\phi}}^a &= \omega^a(\bar{\bs{J}}) + \epsilon \ \avg{f^a}(\bar{\bs{J}}) , \end{align}

while the tilded (fast) variables have the Fourier expansions

\begin{align} \tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \\ \tilde{\phi}^a &= \sum_{\vec{n}\neq \bs{0}} \tilde{\phi}^{a}_{\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \end{align}

where their Fourier coefficients are found from

\begin{align} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) &= \frac{F_{a,\vec{n}}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} , \\ \tilde{\phi}^a_{\vec{n}}(\bar{\bs{J}}) &= \frac{F_{b,\vec{n}}(\bar{\bs{J}})}{(i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}}))^2} \frac{\pd \omega^a}{\pd \bar{J}_b}(\bar{\bs{J}}) + \frac{f^a_\vec{n}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} . \end{align}

References

  1. Fumagalli et al., Nonadiabatic dynamics of eccentric black-hole binaries in post-Newtonian theory, Phys. Rev. D 112, 024012 (2025) ↩ ↩2

  2. Lynch et al., Eccentric self-forced inspirals into a rotating black hole, Class. Quantum Grav. 39 145004 (2022) ↩

  3. Beware: much of the literature uses different conventions for what tilde denotes. I like a bar to denote something average, and a tilde to denote something oscillatory. ↩

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