What is the need for angle-action variables in describing integrable systems?

Question

I am currently reading Chapter 2 of Theoretical Astrophysics Vol 1 by T. Padmanabhan. Here he discusses integrable systems in Hamiltonian mechanics. The idea of angle action variables are introduced as a convenient tool to study integrable Hamiltonian systems.

However I do not see the usefulness of defining these variables as opposed to what I feel are more straightforward ways to solve the problem.

My understanding

Based on my understanding angle-action variables are a set of variables $(\theta_i, I_i)$ such that $I_i$ is a function of the integrals of motion of the system denoted by $F$. More specifically,

$$I_i= \sum_j \int_{\mathcal{C}} p_j(q, F)\ dq_j$$

where the integral is evaluated over some curve $\mathcal{C}$. The nature of this curve is not relevant to my question but is explored in more detail in the text I referred to above. The main point is that the action variables $I_i$ are functions of the integrals of motion $F_j$.

This means that if we make a canonical transformation $(q,p)\to (\theta, I)$ then the new Hamiltonian will only depend on the new momenta $I$ and not on the new coordinates $\theta$. This is because from Hamiltonian equations we have,

$$\frac{\partial H}{\partial \theta_i}= \dot{I_i}= 0$$

where the last step holds because $I_i$ is a function of the integrals of motion and hence are constant.

My understanding is that writing the Hamiltonian only in terms of the momenta result in some pretty neat equations for the evolution of the coordinates $\theta_i$ i.e. they are linear in time. This is what makes angle-action variables useful.

My question

My main question is why do we need to define such a complicated function of the integrals of motion $F$. What if we just made the canonical transformation $(q,p)\to (Q, F)$ where the new momenta are just the integrals of motion?

Even in this scenario the Hamiltonian is just a function of the new momenta. This is because,

$$\frac{\partial H}{\partial Q_i}= \dot{F_i}= 0$$

where the last step holds because $F_i$ is an integral of motion. This method would also give simple equations for the evolution of $Q_i$. What benefits does the angle-action approach provide as opposed to just using the integrals of motion themselves as our new conjugate momenta?

Please let me know if there are any flaws in my understanding that could help resolve my question.

Answer 1

There are several issues, e.g.

1. OP's functions $(F_1,\ldots,F_n)$ are presumably only locally defined in some open neighborhood. To be integrals of motion they should be globally defined, cf. e.g. this & this Phys.SE posts.

2. If an angle variable is periodic, one would like the period to be independent of the action variables. It is unclear that OP's variables $(Q^1,\ldots,Q^n)$ satisfy this.

See also e.g. this related Phys.SE post.

Answer 2

My main question is why do we need to define such a complicated function [...]

OP's question is a fundamental misunderstanding in the first place. If you had instead consulted the standard graduate textbook, Goldstein, the treatment there makes it explicitly clear that action-angle variables are natural, simple, and convenient and should be tried at the earliest possible stages.

To be a bit more useful, let's regurgitate a bit of Goldstein's presentation, paraphrasing only because it is plucked from memory and errors all mine. Consider that, having a simple enough system that is super-integrable, so that the action-angle variables treatment $I_\mu=\frac1{2\pi}\oint p_\mu\mathrm dx^\mu$ (no sum) for all the degrees of freedom, manages to reduce the problem to triviality. Then, for any set of functions, purely of the action variables, that also preserves the necessary degrees of freedom, $f_i=f_i(I_1,I_2,\cdots)$ of arbitrary complexity that is linearly independent (which you may verify, say, by a nowhere-vanishing Wronksian), you could have solved the problem for a different set of angles that these new functions would be corresponding to. After all, these new functions, no matter how crazy, will just be constants under the Hamiltonian flow, because all of their inputs are constants. I mean, by definition, action variables are integrals of motion, and any function purely of just the action variables are thus themselves also integrals of motion.

In particular, one of the two kinds of Hamilton-Jacobi equation solution form, the $W$ kind as opposed to the $S$ kind, is thus necessarily equivalent to action-angle variables; in turn, this means that action-angle variables are immediately useful in helping to solve Hamilton-Jacobi equation. Hamilton-Jacobi equations are said to be the closest approach between classical mechanics and quantum mechanics, and indeed, Schrödinger himself obtained quantum theory by starting from Hamilton-Jacobi equation. Just this alone would already have justified discussing action-angle variables in a general physics course.

But it is far more useful than even that! Of course, if you already know sufficient integrals of motion to completely solve a problem, then, sure, action-angle variables seem to be complicated for no good reason. However, the difficulty is precisely that you do not know the integrals of motion beforehand, and action-angle variables are an explicit prescription for you to search for one. You can start with a system that, although you do not know the integrals of motion yet, but you notice some periodicity, be it of the cyclic type (like vibrations) or open type (like complete rotations), of arbitrary complexity in their periodicity, and use action-angle variables prescription to obtain an integral of motion, basically trivially. You can even obtain it empirically, by studying the long-time-average-of-integral limits, and it would suggest a criterion to look out for, if it exists.

On top of that, action-angle variables come with obvious physical intepretation. Say, in the exactly solvable Keplerian two-body problem, the action-angle variables immediately lead to obvious intepretations, giving rise to angular momentum, Laplace-Runge-Lenz vector, location and direction of the perihelion, etc, and the angle variables are equally easy to physically interpret.

And yet, after all that niceness, we are still not done. When we have no choice but to have a slight perturbation, action-angle variables tend to be secular adiabatic invariants. That is, the action variables are guaranteed to change exceedingly slowly upon slight perturbations, and so are suitable for all sorts of approximation schemes and for stable long-term numerical integration.

So, I have no idea what the OP is misunderstanding when saying that action-angle variables are complicated. They are, instead, way too simple for their utility, and especially their brain-deaded-ness in application. What other comparably performant general tool does not require a tonne of physical insight to wield properly?