Liouville's integrability theorem: action-angle variables

Question

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. 'Remark 11.12' on pg 443 of Fasano-Marmi's 'Analytical Mechanics' suggest that $I_{\alpha}$s can be taken as canonical coordinates.

For a conservative system, the Hamiltonian $H$ is a constant of motion. Let's refer to $H$ as $I_1$. Then $I_1$ becomes one of the canonical momenta. Hence $H$ can be written as $H=I_1$. Application of Hamilton's eqns. of motion implies that only one angle variable $\phi_1$ (corresponding to $I_1$) evolves linearly in time while all others stay constant because $$ \dot{\phi_i}=\frac{\partial H}{\partial I_i} = 0 ~~~~~~~~~~~~~~~~~\mathrm{for~}i\neq1. $$ So, is it true that for every Liouville integrable (described here) and conservative system (where Hamiltonian does not depend on time explicitly), Hamiltonian can be written as a function of only one action variable $I_1$ and only one angle variable (corresponding to $I_1$) evolves linearly in time, whereas others stay constant?

Answer 1

1. Given $n$ functionally independent, Poisson-commuting, globally defined functions $(I_1, \ldots, I_n)$, so that the Hamiltonian $H$ is a function of $(I_1, \ldots, I_n)$ with $\mathrm{d}H\neq 0$, there certainly exist locally defined coordinate transformations: $$ (I_1, I_2,\ldots, I_n)\qquad \longrightarrow \qquad (I^{\prime}_1\!\equiv\!H,I^{\prime}_2, \ldots, I^{\prime}_n). \tag{*}$$ However, without further assumptions, it is not clear whether such globally defined coordinate transformation exists.

2. Moreover, if $(\phi^1,\ldots, \phi^n, I_1, \ldots, I_n)$ are angle-action (AA) variables with a constant (=$I$-independent) period$^1$ matrix $\Pi^{k}_{\ell}$ for the angle variables $(\phi^1,\ldots, \phi^n)$, a coordinate transformation (*) may make the corresponding period matrix [for the new angle variables $(\phi^{\prime 1},\ldots, \phi^{\prime n})$] dependent on the new $(I^{\prime}_1, \ldots, I^{\prime}_n)$ variables.

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$^1$ For the $n$-torus.

Answer 2

In the Hamiton-Jacobi (HJ) approach, the Hamiltonian does not stay the same. It changes via (Eq. 9.17-c of Goldstein)

$$ K = H + \frac{\partial F_2}{\partial t}, $$ where $K$ is the transformed Hamiltonian. In HJ approach, we tune $F_2$ in such a way that $K=0$ (Eq. 10.2 of Goldstein). The above question assumes that $K=H$ which is possible only if $F_2$ is independent of $t$ which is not what happens in the HJ theory.

Answer 3

I believe I understand your question. I think Different action-angle variables for a 2D harmonic oscillator is a good example. The 2D oscillator is

$$H = H_1 = \frac12( p_x^2 + p_y^2 + x^2 + y^2)$$

which may be split into $H = H_x + H_y$

where

$$H_x = \frac12(p_x^2 + x^2)$$

$$H_y = \frac12(p_y^2 + y^2)$$

and then you have one hamiltonian, but you have replaced it with two action variables and it may be solved to obtain two angle variables, $\phi_{H_x}$ and $\phi_{H_y}$, each with constant rates.

I hope that helps.

I think we also need an answer to address the following, which I believe is at the heart of the original question: Suppose I wanted my action variables to be $H$ and $H_x$. When I take the partial derivative of the Hamiltonian ($H=H_1$) with respect to $H_1$, I get 1 so $\phi_{H_1}$ has a constant rate. When I take the partial derivative of the Hamiltonian ($H=H_1$) with respect to $H_x$, I get zero (0), so $\phi_{H_x}$ is constant? Why is this wrong?