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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 function $F_1, \ldots, F_n$ from Definition 3 are automatically Poisson-commuting, 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.

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.

I) 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 $I_1, \ldots, I_n: {\cal M}\to \mathbb{R}$, so that the Hamiltonian $H$ is a function of $I_1, \ldots, I_n$, only. See also this related Phys.SE post.

3. Definition. The system is (completely) $H$-separable if the Hamiltonian $H$ is (completely) separable in an atlas of Darboux coordinates charts.

4. Theorem. Integrability $\Rightarrow$ $H$-separability. Proof: Use Caratheodory-Jacobi-Lie theorem to extend the Poisson-commuting coordinates $(I_1, \ldots, I_n)$ into an atlas of Darboux coordinate neighborhoods. The Hamiltonian $H(I)$ is then on separable form. $\Box$

5. Definition. The system is called (completely) $W$-separable if Hamilton's characteristic function $W$ is of the form $$ W(q,\alpha)~=~ \sum_{k=1}^nW _k(q^k,\alpha),\tag{1}$$ where $\alpha=(\alpha_1,\ldots,\alpha_n)$ are $n$ independent integration constants, and the coordinates $(q^1,\ldots, q^n,p_1,\ldots,p_n)$ are Darboux coordinates, where $$ p_k~:=~\frac{\partial W}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{2}$$ Globally with an atlas of such Darboux coordinate charts, the phase space ${\cal M}$ should consist of $n$ linearly independent 1-dimensional foliations (sharing the $\alpha$-coordinates).

6. $W$-separability implies $H$-separability, but the opposite need not be the case, as there is no guarantee that a globally defined Hamilton's characteristic function $W$ exists as a solution to Hamilton-Jacobi equation.

7. 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 $\omega=\mathrm{d}J_k\wedge \mathrm{d}w^k$ 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 $w^k\sim w^k+1$.)

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

9. Case where $W$-separability $\Rightarrow$ integrability: Assume that the $n$ integration constants $\alpha=(\alpha_1,\ldots,\alpha_n)$ can be identified with functionally independent, Poisson-commuting, constants of motion $I_k(z)$, $k\in\{1, \ldots, n\}$.

10. Case where integrability $\Rightarrow$ AA-property: Assume $\forall a=(a_1,\ldots, a_n)\in\mathbb{R}^n$ that the level sets $\bigcap_{k=1}^n I_k^{-1}(\{a_k\})$ 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.

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 function $F_1, \ldots, F_n$ from Definition 3 are automatically Poisson-commuting, 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.

I) 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 $I_1, \ldots, I_n: {\cal M}\to \mathbb{R}$, so that the Hamiltonian $H$ is a function of $I_1, \ldots, I_n$, only. See also this related Phys.SE post.

3. Definition. The system is (completely) $H$-separable if the Hamiltonian $H$ is (completely) separable in an atlas of Darboux coordinates charts.

4. Theorem. Integrability $\Rightarrow$ $H$-separability. Proof: Use Caratheodory-Jacobi-Lie theorem to extend the Poisson-commuting coordinates $(I_1, \ldots, I_n)$ into an atlas of Darboux coordinate neighborhoods. The Hamiltonian $H(I)$ is then on separable form. $\Box$

5. Definition. The system is called (completely) $W$-separable if Hamilton's characteristic function $W$ is of the form $$ W(q,\alpha)~=~ \sum_{k=1}^nW _k(q^k,\alpha),\tag{1}$$ where $\alpha=(\alpha_1,\ldots,\alpha_n)$ are $n$ independent integration constants, and the coordinates $(q^1,\ldots, q^n,p_1,\ldots,p_n)$ are Darboux coordinates, where $$ p_k~:=~\frac{\partial W}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{2}$$ Globally with an atlas of such Darboux coordinate charts, the phase space ${\cal M}$ should consist of $n$ linearly independent 1-dimensional foliations (sharing the $\alpha$-coordinates).

6. $W$-separability implies $H$-separability, but the opposite need not be the case, as there is no guarantee that a globally defined Hamilton's characteristic function $W$ exists as a solution to Hamilton-Jacobi equation.

7. Definition. The system has the AA-property if there exists an atlas of angle-action coordinates $(w^1,\ldots, w^n,J_1,\ldots, J_n)$ with Hamilton's characteristic function on a separable form $W=\sum_{k=1}^nw^kJ_k$, with symplectic form $\omega=\mathrm{d}J_k\wedge \mathrm{d}w^k$ 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 $w^k\sim w^k+1$.)

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

9. Case where $W$-separability $\Rightarrow$ integrability: Assume that the $n$ integration constants $\alpha=(\alpha_1,\ldots,\alpha_n)$ can be identified with functionally independent, Poisson-commuting, constants of motion $I_k(z)$, $k\in\{1, \ldots, n\}$.

10. Case where integrability $\Rightarrow$ AA-property: Assume $\forall a=(a_1,\ldots, a_n)\in\mathbb{R}^n$ that the level sets $\bigcap_{k=1}^n I_k^{-1}(\{a_k\})$ 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.

I) 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 $I_1, \ldots, I_n: {\cal M}\to \mathbb{R}$, so that the Hamiltonian $H$ is a function of $I_1, \ldots, I_n$, only. See also this related Phys.SE post.

3. Definition. The system is (completely) $H$-separable if the Hamiltonian $H$ is (completely) separable in an atlas of Darboux coordinates charts.

4. Theorem. Integrability $\Rightarrow$ $H$-separability. Proof: Use Caratheodory-Jacobi-Lie theorem to extend the Poisson-commuting coordinates $(I_1, \ldots, I_n)$ into an atlas of Darboux coordinate neighborhoods. The Hamiltonian $H(I)$ is then on separable form. $\Box$

5. Definition. The system is called (completely) $W$-separable if Hamilton's characteristic function $W$ is of the form $$ W(q,\alpha)~=~ \sum_{k=1}^nW _k(q^k,\alpha),\tag{1}$$ where $\alpha=(\alpha_1,\ldots,\alpha_n)$ are $n$ independent integration constants, and the coordinates $(q^1,\ldots, q^n,p_1,\ldots,p_n)$ are Darboux coordinates, where $$ p_k~:=~\frac{\partial W}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{2}$$ Globally with an atlas of such Darboux coordinate charts, the phase space ${\cal M}$ should consist of $n$ linearly independent 1-dimensional foliations (sharing the $\alpha$-coordinates).

6. $W$-separability implies $H$-separability, but the opposite need not be the case, as there is no guarantee that a globally defined Hamilton's characteristic function $W$ exists as a solution to Hamilton-Jacobi equation.

7. 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 $\omega=\mathrm{d}J_k\wedge \mathrm{d}w^k$ 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 $w^k\sim w^k+1$.)

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

9. Case where $W$-separability $\Rightarrow$ integrability: Assume that the $n$ integration constants $\alpha=(\alpha_1,\ldots,\alpha_n)$ can be identified with functionally independent, Poisson-commuting, constants of motion $I_k(z)$, $k\in\{1, \ldots, n\}$.

10. Case where integrability $\Rightarrow$ AA-property: Assume $\forall a=(a_1,\ldots, a_n)\in\mathbb{R}^n$ that the level sets $\bigcap_{k=1}^n I_k^{-1}(\{a_k\})$ 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.