4
$\begingroup$

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 :/

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – Qmechanic
    Commented Feb 17, 2018 at 16:11

2 Answers 2

8
$\begingroup$

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.

--

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

$\endgroup$
0
0
$\begingroup$

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 $\left\{\phi_m,F_i\right\}= \mathrm{constant}(F_k) \,$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.