I am asked to show that, a system is completely integrable Liouville if and only if its Hamilton-Jacobi equation is completely separable. I get the idea and understand that is very related to the Action-Angle coordinates and one of Liouville's theorems. Two texts have been of help: Jose & Saletan and Arnol'd.

I see how separability leads to AA coordinates and thus to an integrable solution of the system, but why is it an "if and only if" relationship? Why can't there exist systems with non-separable HJ, but reducible to integrals?


In this answer, we elaborate on the various definitions of integrability, separability & AA-property, in order to expose their (slight) differences.

  1. Let there be given a finite-dimensional autonomous Hamiltonian system, defined on a connected $2n$-dimensional symplectic manifold $({\cal M},\{\cdot ,\cdot \})$.

  2. Definition. The system is (completely) Liouville integrable if there exist $n$ functionally independent, Poisson-commuting, globally defined functions $F_1, \ldots, F_n: {\cal M}\to \mathbb{R}$, so that the Hamiltonian $H=H(F)$ is a function of $F_1, \ldots, F_n$, only. See also this related Phys.SE post.

  3. Definition. The system is (completely) $H$-separable if there exists an atlas of Darboux coordinates $ q^1, \ldots, q^n, p_1, \ldots, p_n : {\cal U}\to \mathbb{R}$ with separation functions $F_1, \ldots, F_n: {\cal U}\to \mathbb{R}$ on triangular form $$F_1~=~F_1(q^1,p_1), \qquad F_2~=~F_2(q^2,p_2; F_1), \qquad F_3~=~F_3(q^3,p_3; F_1,F_2), \tag{1}$$ $$\qquad \ldots, \qquad F_n~=~F_n(q^n,p_n; F_1,\ldots, F_{n-1}), $$
    such that the Hamiltonian $H=H(F)$ is a function of $F_1, \ldots, F_n$, only.

  4. Note that the separation functions $F_1, \ldots, F_n$ from Definition 3 are automatically Poisson-commuting and constants of motion, but not necessarily functionally independent. A globally defined $H$-separating Darboux coordinate system with functionally independent separation functions implies integrability.

  5. Theorem. Integrability $\Rightarrow$ $H$-separability. Proof: Use Caratheodory-Jacobi-Lie theorem to extend the Poisson-commuting coordinates $(F_1, \ldots, F_n)$ into an atlas of Darboux coordinate neighborhoods. The Hamiltonian $H(F)$ is then on separable form. $\Box$

  6. Definition. The system is called (completely) $W$-separable if there exists an atlas of Darboux coordinates $ q^1, \ldots, q^n, p_1, \ldots, p_n : {\cal U}\to \mathbb{R}$ and a Hamilton's characteristic function $W: {\cal U}\times \mathbb{R}^n\to \mathbb{R}$ of the form $$ W(q;\alpha)~=~ \sum_{k=1}^n W_k(q^k;\alpha_1, \ldots, \alpha_n),\tag{2}$$ where $\alpha=(\alpha_1,\ldots,\alpha_n)$ are $n$ independent integration constants, and where $$ p_k~:=~\frac{\partial W}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{3}$$ such that the Hamilton-Jacobi (HJ) equation $$ H\left(q,\frac{\partial W(q;\alpha)}{\partial q}\right)~=~h(\alpha)\tag{4} $$ is satisfied. Here $h:\mathbb{R}^n\to \mathbb{R}$ is a given function.

  7. Case where $W$-separability $\Rightarrow$ $H$-separability: Assume that the $n$ integration constants $\alpha=(\alpha_1,\ldots,\alpha_n)$ can be identified with Poisson-commuting separation functions $F_k(z)$, $k\in\{1, \ldots, n\}$. Then $H=h(F)$ and the separation functions become constants of motion.

  8. $H$-separability does not necessarily imply $W$-separability as there is no guarantee that a globally defined Hamilton's characteristic function $W$ exists as a solution to the HJ equation.

  9. Definition. The system has the AA-property if there exists an atlas of angle-action coordinates $(w^1,\ldots, w^n,J_1,\ldots, J_n)$, where the symplectic $2$-form $\omega=\sum_{k=1}^n\mathrm{d}J_k\wedge \mathrm{d}w^k$ is on Darboux form, where each AA-coordinate system is $w$-complete, and where the Hamiltonian $H=H(J)$ does not depend on the angles $w^k$. (We allow non-compact "angle" variables. The compact angle variables has unit period $w^k\sim w^k+1$.)

  10. The AA-property clearly implies all the separability conditions. A globally defined angle-action coordinate system implies integrability.

  11. Case where integrability $\Rightarrow$ AA-property: Assume $\forall f=(f_1,\ldots, f_n)\in\mathbb{R}^n$ that the level sets $\bigcap_{k=1}^n F_k^{-1}(\{f_k\})$ are compact in ${\cal M}$. Then the Liouville-Arnold theorem shows the AA-property. For a proof of the Liouville-Arnold theorem, see my Phys.SE answer here.

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    $\begingroup$ Note for later: J.V. Jose & E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998, write p. 291: There also exist dynamical systems whose HJ equations cannot be separated, but which can nevertheless be integrated in other ways (see Perelomov, 1990). P.321: ...there exist dynamical systems that are not separable on $\mathbb{Q}$ but are completely integrable in the sense of the LI theorem (see Toda, 1987). $\endgroup$ – Qmechanic Apr 8 '18 at 13:46
  • $\begingroup$ Great answer as usual...could I just asky why in the point 7 it is said that the saparation functions $F_k$ "become constants of motion", if they already are by point 4? $\endgroup$ – Lo Scrondo Jun 6 at 16:07

I think It is not clear to what extent the notions of integrability and separability imply each other. Strict proof of integrability requires finding the coordinate transformations that give the action-angle variables and verifying that they are smooth and invertible. On the other hand, it is commonly stated that a system with $n$ degrees of freedom is Liouville integrable if the dynamics is able to produce $n$ periods of motion. In many systems symmetries are obvious in the sense that they provide constants of motion and this knowledge helps you to get the action-angle coordinates. But this may not be always possible even if the system under consideration have bounded orbits: you may encounter cases where trajectories are quasiperiodic but there is no guarantee that a coordinate transformation to action-angle coordinate exists. A similar situation arises in the quantum version of the Toda lattice. You may take a look at brief discussion about this on Gutzwiller's Chaos in Classical and Quantum mechanics, section 3.7


Jose-Saletan on pg 321 clearly states that action-angle variables may exist despite the absence of separability of the Hamilton-Jacobi equation. So, I think the use of the phrase "if and only if" is wrong.


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