# On degenerate integrable hamiltonian systems

Is there some reference where the existence of local generalized action-angle variables is discussed in some detail for concrete examples of hamiltonian systems of mechanical type?

After Dazord and Delzant, by local generalized action-angle coordinates on a symplectic manifold $(M^{2n},\omega)$ I mean a locally trivial bundle $\pi:M\to P$ with fiber $\mathbb{T}^k$ and having a trivializing atlas whose elements $(U,\phi:\pi^{-1}(U)\to U\times\mathbb{T}^k)$ satisfy the following property:
$\phi_{\ast}\omega=\sum_{i=1}^k dJ_i\wedge \theta_i+\sum_{i=1}^{n-k}dp_i\wedge dp_i$ where $J_1,\ldots,J_k,p_1,\ldots,p_{n-k},q_1,\ldots,q_{n-k}$ are adapted coordinates on $U$ and $\theta_1\ldots,\theta_k$ is a base of invariant $1$-forms on $\mathbb{T}^k$.

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How about the book "Global Aspects of Classic Integrable Systems" by Cushman and Bates? it is concerned with monodromy and the global existence of action-angle coordinates, so I'm pretty sure it does a good job of sketching the local picture, although I don't have a copy with me right now. –  jvkersch Jun 21 '11 at 22:29
Dear jvkersch, thank you for the reference. Even if in this book there is no mention of degenerate (or super, non commutative) integrability, the detailed analysis of the topology of the momentum map for many classical ham. systems ( Euler top, kepler problem, harmonic oscillator,...; that, by the way, are superintegrable) should permit to recognize the existence of generalized action-angle coordinates at least on a open subset of the phase space. –  Giuseppe Tortorella Jun 23 '11 at 6:32

If I understand correctly what you are after, a good place to have a look at is this expository article by Fasso'

F. Fasso'. Superintegrable Hamiltonian systems: geometry and perturbations. Acta Appl. Math. 87 (2005),

in which he reviews the case of superintegrable Hamiltonian systems defined by Mischenko and Fomenko (this is the mechanical case that you should be after). Some examples are briefly discussed and there are references to longer expositions (e.g. the Euler-Poinsot top).

Just a final word of warning. In the completely integrable Hamiltonian systems literature, "degenerate" refers to the case when the underlying system is completely integrable a la Liouville, but there are singularities which are not, in some sense, of Morse-Bott type. If I am not mistaken, the case that you discuss here is often referred to as "superintegrable" or "non-commutatively integrable", as defined by Mischenko and Fomenko (and others).

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Dear Daniele Sepe, thanks for your reference. In this paper the integrability condition of Mischenko and Fomenko is interpreted as sufficient for the existence of an isotropic and symplectically complete fibration (FISC after Dazord and Delzant) and hence of generalized action-angle coordinates. I learn also that the first use of the notion of FISC in concrete examples of mechanical interest is the book Nonlinear Poisson Bracket of Maslov and Karasev, and the other paper of Fassò on Euler-Poinsot that you cite. Thank you. –  Giuseppe Tortorella Jun 23 '11 at 17:26