# Liouville's integrability theorem: action-angle variables

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?

• In a system with $n$ degrees of freedom, there are $n$ mutually commuting constants of motion, $I_{1}=H,...,I_{n}$ where $\{I_{i},I_{j}\}=0$ for $i,j=1,...,n$. In other words, there are $n$ constant frequencies - which means the $q_{i}$ are periodic in time. In your case I presume $n=1$? – Cinaed Simson May 19 '19 at 7:41

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.

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.

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?