# Proof of constructing action-angle coordinates on Hamiltonian system

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures. I am looking at a proof of the construction of these coordinates, and I am not certain of a certain part. On page 180 of Mathematical Aspects of Celestial Mechanics it says that the Poisson bracket $$\{F_i,\varphi_j\}$$ is constant on $$M_f$$, the Lagrangian tori. I tried to prove it but I am not sure of the proof. Here is my attempt:

Choose Darboux coordinates $$\{\bf{p},\bf{q}\}$$ and by the coordinate representation of the Poisson bracket, we have

$$\{F_i, \varphi_j\} = \sum\limits_{k=1}^n \dfrac{\partial F_i}{\partial p_k} \dfrac{\partial \varphi_j}{\partial q_k} - \dfrac{\partial F_i}{\partial q_k} \dfrac{\partial \varphi_j}{\partial p_k} = \omega_j\sum\limits_{k=1}^n \dfrac{\partial F_i}{\partial p_k} \bigg(\dfrac{\partial F_1}{\partial p_k}\bigg)^{-1} - \dfrac{\partial F_i}{\partial q_k} \bigg(\dfrac{\partial F_1}{\partial q_k}\bigg)^{-1} = 2\omega_j,$$

where the second equality is because of the Hamiltonian equations (with $$H = F_1$$) and the fact that $$\dfrac{\partial \varphi_j}{\partial t} = \omega_j$$ where the $$\varphi_j$$ are the angle coordinates that are described by linear flow on $$M_f$$, and the third equality is because of $$\dfrac{\partial F_i}{\partial F_1} = \delta_{i1}$$. However, I am not so sure of the third equality because it just feels odd to differentiate the $$F_i$$'s by the $$F_1$$'s.

Any form of help will be appreciated as I am doing my thesis now and struggling :/

• Minor comment to the post (v2): Note that Arnold's book as well as Wikipedia use the opposite sign convention than OP for the Poisson bracket. Commented Feb 17, 2018 at 16:11

OP's actual question follows directly from Theorem 5.3 in Ref. 2, but that leaves the obvious question: How to prove Theorem 5.3? It seems the only really satisfying answer would be to outline a complete proof of the Liouville-Arnold Theorem. This is what we intend to do in this answer.

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

Definition 1. 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 $$H$$ is a function of $$F_1, \ldots, F_n$$, only. In other words, the functions $$F_1, \ldots, F_n$$ are integrals of motion. See also this related Phys.SE post.

Definition 2. 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 $$w^k\sim w^k+1$$.)

Liouville-Arnold Theorem. Given a completely Liouville integrable system, and assume $$\forall f=(f_1,\ldots, f_n)\in\mathbb{R}^n$$ that the level sets $${\cal M}_f:=\bigcap_{k=1}^n F_k^{-1}(\{f_k\})$$ are compact in $${\cal M}$$. Then each connected components of the level sets are $$n$$-tori, and the system has the AA-property.

Sketched proof:

1. On one hand, from the commuting Hamiltonian vector fields $$X_k:=-\{F_k, \cdot\}$$, we generate an abelian flow \begin{align} g(t)~:=~& \exp(\sum_{k=1}^n t^k X_k), \cr g(t)g(t^{\prime})~=~&g(t+t^{\prime}), \qquad t,t^{\prime}~\in~\mathbb{R}^n .\end{align} \tag{1} We focus on a connected component of a compact level-set $${\cal M}_f$$. We conclude from the compactness assumption that the $$t$$-parameters generate a periodic torus action $$t^k\sim t^k+\Pi^k_{\ell}(F)$$ (for each connected component), where $$\Pi^k_{\ell}(F)$$ is a period matrix.

2. On the other hand, from Frobenius theorem for the involutive distribution $$\Delta:={\rm span}_{\mathbb{R}}(X_1,\ldots,X_n)$$, there exists an $$n$$-foliation. In other words, there exists an atlas of local coordinates of the form $$(\varphi^1, \ldots,\varphi^1, G_1, \ldots, G_n)$$, where firstly the $$\varphi^k$$-coordinates parametrize the leaves, and secondly the $$G_k$$-coordinates (like the $$F_{\ell}$$-coordinates) label the leaves. Hence the $$G_k$$-coordinates must be functions of $$F_{\ell}$$-coordinates. Two overlapping local coordinates systems (say, primed and unprimed) are related via \begin{align} \varphi^{\prime k} ~=~& \varphi^{\prime k}(\varphi,G), \cr G^{\prime}_k ~=~&G^{\prime}_k(G), \cr k~\in~&\{1, \ldots, n\}.\end{align}\tag{2} We may w.l.o.g. assume that the local $$\varphi^k$$-coordinates always stratify the Hamiltonian vector fields $$\frac{\partial}{\partial \varphi^k}~=~X_k, \qquad k~\in~\{1, \ldots, n\}.\tag{3}$$ Notice that the abelian flow then becomes $$g(t)~\stackrel{(1)+(3)}{=}~ \exp(\sum_{k=1}^n t^k \frac{\partial}{\partial \varphi^k}) , \qquad t~\in~\mathbb{R}^n .\tag{4}$$ It becomes clear that we then should identify the parameter $$t^k$$ with the coordinate $$\varphi^k$$. Eqs. (2) & (3) narrow coordinate transformations down to the form $$\varphi^{\prime k} = \varphi^k+ h^k(F)$$. It becomes clear that we can take unions of such local coordinate patches to create a $$\varphi$$-complete coordinate system (for each connected component) with a periodicity condition $$\varphi^k\sim \varphi^k+\Pi^k_{\ell}(F)$$. In other words, the $$\varphi^k$$ are un-normalized angle variables.

3. Given a connected component of a compact level set, which must be an $$n$$-torus $$\cong (\mathbb{S}^1)^n$$, we can find a tubular neighborhood $${\cal U}\subseteq {\cal M}$$ with coordinates $$(\varphi^1,\ldots,\varphi^n,F_1, \ldots, F_n)$$, where the $$F$$-coordinates live in some contractible space, say, an $$n$$-box. We now restrict the construction to this tubular neighborhood $${\cal U}$$.

The fundamental Poisson brackets are not necessarily on Darboux form$$^1$$ \begin{align} \{ \varphi^k, \varphi^{\ell}\} ~=~& \beta^{k\ell}, \cr \{ \varphi^k, F_{\ell}\}~=~& \delta^k_{\ell}, \cr \{ F_k, F_{\ell}\}~=~& 0, \cr k,\ell ~\in~&\{1, \ldots, n\}.\end{align}\tag{5} The corresponding symplectic form is the inverse structure \begin{align}\omega~=~&\sum_{k=1}^n\mathrm{d}F_k \wedge \mathrm{d}\varphi^k + \beta, \cr \beta~:=~& \frac{1}{2} \sum_{k,\ell=1}^n \mathrm{d}F_k ~\beta^{k\ell} \wedge \mathrm{d}F_{\ell}.\end{align} \tag{6} However, there is a bi-grading of two anti-commuting differentials \begin{align}\mathrm{d}~=~&\mathrm{d}^{(\varphi)}+ \mathrm{d}^{(F)}, \cr \mathrm{d}^{(\varphi)}~:=~&\sum_{k=1}^n\mathrm{d}\varphi^k \frac{\partial}{\partial \varphi^k},\cr \mathrm{d}^{(F)}~:=~&\sum_{k=1}^n\mathrm{d}F_k \frac{\partial}{\partial F_k} .\end{align} \tag{7} The closedness $$\mathrm{d}\beta=0$$ implies firstly $$\mathrm{d}^{(\varphi)}\beta=0$$ (i.e. $$\beta^{k\ell}$$ can not depend on $$\varphi$$.), and secondly $$\mathrm{d}^{(F)}\beta=0$$. By Poincare Lemma there exists a $$(0,1)$$-form $$\alpha~=~\sum_{k=1}^n\alpha^k(F)~\mathrm{d}F_k\tag{8}$$ on $${\cal U}$$ such that the symplectic $$(0,2)$$-form $$\beta|_{\cal U}~=~\mathrm{d}^{(F)}\alpha~=~\mathrm{d}\alpha.\tag{9}$$

Define new coordinates \begin{align} q^k~:=~\varphi^k -\alpha^k(F)~\sim~& q^k+\Pi^k_{\ell}(F), \cr k,\ell ~\in~&\{1, \ldots, n\},\end{align}\tag{10}
which are periodic with the same periods. It is easy to check that $$(q^1,\ldots,q^n,F_1, \ldots, F_n)$$ is a $$q$$-complete Darboux coordinate neighborhood $${\cal U}$$ with a symplectic potential $$(1,0)$$-form \begin{align}\vartheta~=~&\sum_{k=1}^nF_k ~\mathrm{d}q^k, \cr \mathrm{d}\vartheta~=~&\sum_{k=1}^n\mathrm{d}F_k \wedge \mathrm{d}q^k ~=~\omega|_{\cal U}.\end{align}\tag{11}

4. At this point we take a step backwards. (This is not needed for the AA existence proof, but in practice it is useful to outline the construction for more general coordinates. See also remark 7 below.) Let us more generally imagine that we have a tubular neighborhood $${\cal U}\subseteq {\cal M}$$ with some $$q$$-complete (not necessarily Darboux) coordinates $$(q^1,\ldots,q^n,F_1, \ldots, F_n)$$ with a symplectic potential $$(1,0)$$-form $$\vartheta~=~\sum_{k=1}^n p_k(q,F) ~\mathrm{d}q^k,\tag{12}$$ such that $$q^k\sim q^k+\Pi^k_{\ell}(F)$$.

Consistency with the non-degenerate symplectic 2-form \begin{align}\omega|_{\cal U}~=~& \mathrm{d}\vartheta\cr ~\stackrel{(12)}{=}~& \underbrace{\sum_{k,\ell=1}^n\mathrm{d}q^k ~\frac{\partial p_{\ell}}{\partial q^k} \wedge \mathrm{d}q^{\ell}}_{=0\text{ because }\{ F_k, F_{\ell}\}=0 } +\sum_{k,\ell=1}^n\mathrm{d}F_k ~\frac{\partial p_{\ell}}{\partial F_k} \wedge \mathrm{d}q^{\ell},\end{align}\tag{13} implies:

• that $$\{q^k,q^{\ell}\}=0$$.

• the Maxwell relations $$\frac{\partial p_{\ell}(q,F)}{\partial q^k} ~=~ (k\leftrightarrow \ell) , \qquad k,\ell~\in~\{1, \ldots, n\}.\tag{14}$$ Eq. (14) is an integrability condition for the local existence of the Hamilton's characteristic function $$W(q,F)$$, cf. Section 6 below.

• that the matrix $$\frac{\partial p_{\ell}}{\partial F_k}$$ is invertible. After possibly restricting to a smaller tubular neighborhood $${\cal V}\subseteq {\cal U}$$, this proves that $$(q^1,\ldots,q^n,p_1, \ldots, p_n)$$ are Darboux coordinates on $${\cal V}$$. (Here we have used the inverse function theorem.)

5. Define action variables $$J_k(q_{\ast},F)~:=~ \oint_{C_k(q_{\ast})}\! \left. \vartheta \right|_{\text{fixed } F}, \qquad k~\in~\{1, \ldots, n\}, \tag{15}$$ where $$C_k(q_{\ast})$$ are a $$k$$'th 1-cycle of the tori in the $$q$$-space $$M$$ (aka. configuration space) that starts and ends at the same point $$q_{\ast}\in M$$. Use Stokes' theorem \begin{align} J_k(q^{\prime}_{\ast},F)-J_k(q_{\ast},F)~:=~& \oint_{\partial A}\! \left.\vartheta \right|_{\text{fixed } F}\cr ~=~& \iint_A\! \left. \omega \right|_{\text{fixed } F}~=~0, \cr k~\in~&\{1, \ldots, n\}, \end{align}\tag{16} to show that $$J_k(q_{\ast},F)$$ do not depend on $$q_{\ast}$$.

At this point we assume that the matrix $$\frac{\partial J_k}{\partial F_{\ell}}$$ is non-degenerate. (This is satisfied for the important case where $$p_k(q,F)=F_k$$.) After possibly restricting to a smaller tubular neighborhood $${\cal V}\subseteq {\cal U}$$, this proves that $$(q^1,\ldots,q^n,J_1, \ldots, J_n)$$ are coordinates on $${\cal V}$$. (Here we have used the inverse function theorem.)

6. Define Hamilton's characteristic function $$W(\gamma,F)~:=~ \int_{\gamma}\!\left. \vartheta \right|_{\text{fixed } F}, \tag{17}$$ where $$\gamma \subseteq M$$ is an oriented curve in the $$q$$-space $$M$$ from a fiducial point $$q_{\ast}(F)$$ to an endpoint, which we (with a slight abuse of notation) denote $$q$$. One may again show that the definition (17) only depend on the homotopy class (i.e. winding numbers) of $$\gamma$$. Hence we may rewrite definition (17) in a slight different notation as $$W(q,F)~:=~ \int_{q_{\ast}}^q\!\left. \vartheta \right|_{\text{fixed } F}, \tag{18}$$ where we use the multi-valued nature of the $$q$$-coordinate system to indicate the winding number. Note that $$p_k(q,F)~=~\frac{\partial W(q,F)}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{19}$$ Define $$W(q,J)~:=~W(q,F(J))$$. Similarly, $$p_k(q,J)~=~\frac{\partial W(q,J)}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{20}$$ Then if we shift with a period \begin{align} \Delta_k W ~:=~&W(q+\Pi_k,J)-W(q,J)~=~J_k, \cr k~\in~&\{1, \ldots, n\}.\end{align} \tag{21} Define new angle variables $$w^k(q,J)~:=~\frac{\partial W(q,J)}{\partial J_k}, \qquad k~\in~\{1, \ldots, n\}. \tag{22}$$
In other words, $$W$$ is a type-2 generating function for the canonical transformation $$(q,p)\to (w,J)$$. Then the periods have unit-length \begin{align} \Delta_{\ell} w^k~:=~& w^k(q+\Pi_{\ell},J)-w^k(q,J)~=~ \delta^k_{\ell}, \cr k,\ell~\in~&\{1, \ldots, n\}. \end{align}\tag{23} $$\Box$$

7. Remark. If a tubular neighborhood $${\cal U}\subseteq {\cal M}$$ of a level set torus is covered by a single Darboux coordinate chart $$(q^1,\ldots,q^n,p_1,\ldots,p_n)$$ such that the projection of a level set torus to the $$q$$-space $$M$$ only contain isolated turning points, then it is still possible to define action variables (15), Hamilton's characteristic function (17), etc, via appropriate generalizations. E.g. the analogue of eq. (17) is $$W(\gamma,F)~:=~ \int_{\sigma_F\circ \gamma}\!\left. \vartheta \right|_{\text{fixed } F}, \tag{24}$$ where $$\sigma_F$$ denotes the $$F$$-dependent lift from $$q$$-space $$M$$ to the level set torus.

8. Example: The simple harmonic oscillator. The Hamiltonian $$H~=~\frac{p^2}{2} + \frac{q^2}{2}\tag{25}$$ is an integral of motion. The system is integrable in the 2D phase space (except for the origin, which is a singular orbit). We can define an angle variable \begin{align} \varphi~:=~&{\rm atan2}(q,p), \cr p+iq~=~&\sqrt{2H}e^{i\varphi}, \cr q~=~&\sqrt{2H}\sin\varphi, \cr p~=~&\sqrt{2H}\cos\varphi\cr ~=~&\pm \sqrt{2H-q^2}.\end{align}\tag{26} The action variable reads \begin{align} J~\stackrel{(15)}{=}~&\oint \!\left. p ~\mathrm{d}q \right|_{\text{fixed } H}\cr ~=~&2H\int_0^{2\pi}\! \mathrm{d}\varphi~\cos^2\varphi\cr ~=~&2\pi H.\end{align}\tag{27}
Hamilton's characteristic function reads \begin{align}W(q,H)~\stackrel{(18)}{=}~&\int_{q_{\ast}}^q \!\left. p ~\mathrm{d}q \right|_{\text{fixed } H}\cr ~=~&2H\int_0^{\varphi(q,H)}\! \mathrm{d}\varphi~\cos^2\varphi\cr ~=~&H\left. \{\varphi + \frac{1}{2}\sin 2 \varphi \}\right|_{\varphi=\varphi(q,H)}\cr ~=~&\left. \{ H~{\rm atan2}(q,p) + \frac{1}{2} q p\}\right|_{p=p(q,H)}.\end{align}\tag{28} The angle variable becomes \begin{align}2\pi w~\stackrel{(22)}{=}~& 2\pi\frac{\partial W(q,J)}{\partial J}\cr ~\stackrel{(27)}{=}~&\frac{\partial W(q,H)}{\partial H}\cr ~\stackrel{(28)}{=}~& {\rm atan2}(q,p) + \left\{H\frac{1}{1+(q/p)^2} \frac{-q}{p^2}+\frac{q}{2}\right\}\frac{\partial p(q,H)}{\partial H}\cr ~=~&{\rm atan2}(q,p)\cr ~=~&\varphi,\end{align} \tag{29}
as expected.

References:

1. V.I. Arnold, Mathematical Methods of Classical Mechanics, 1989; $$\S$$49-50.

2. V.I. Arnold, V.V. Kozlov, & A.I. Neishtadt, Mathematical Aspects of Celestial Mechanics, 2006; Subsections 5.1.1-2 & 5.2.1.

3. J.H. Lowenstein, Essentials of Hamiltonian Dynamics, 2012; Sections 3.1-2 & 3.5.

4. M. Taylor, Partial Differential Equations, Basic Theory, 2011; $$\S$$16.

5. J.V. Jose & E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998; Subsection 6.2.2.

6. A. Fasano & S. Marmi, Analytic Mechanics, 2006; Sections 11.5 + 11.6.

7. N.A. Lemos, Analytic Mechanics, 2018; Section 9.7.

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$$^1$$ Eq. (5) proves OP's actual question, in the sense that such coordinates exist.

It seems to me that they are making the identification $F_i = I_i = y_i$ where the $y_i$ are the ones discussed in Theorem 5.3 (page 174) and, as they mention, they assume the hypotheses of Theorem 5.3 to hold. As a consequence, they say that Hamilton's equations with respect to all $F_i$ have the form $\dot{\phi}_m=\left\{\phi_m,F_i\right\}= \mathrm{constant}$ as well as $\dot{y}_m=\left\{y_m,F_i\right\}= \mathrm{constant}$. In particular, with the choice $y_s=F_s$, we have by hypothesis $0=\left\{F_m,F_i\right\}=\dot{F}_m$. The $F_m$ are then independent constants pariwise in involution. Hence we have $\left\{\phi_m,F_i\right\}= \mathrm{constant}(F_k) \,$.