Integer lattices of action-angle variables for ``spherical pendulum'' system (Q2018035)

The paper considers the well-studied spherical pendulum and proceeds to reprove several results that are known for this system, thus providing a self-contained exposition of the properties of this system. It focuses on the construction of the action integer lattice and its use for the computation of the monodromy matrix and the topology of energy levels. The study of the spherical pendulum goes back to \textit{C. Huygens} [``L'horloge à pendule'', in: Œuvres complètes 18, 69--368 (1673)]. The system has recently received renewed attention as it was the first integrable Hamiltonian system (the integrals are the energy \(E\) and the vertical component of the angular momentum \(M_z\)) that was shown to have Hamiltonian monodromy [\textit{J. J. Duistermaat}, Comm. Pure Appl. Math. 33, 687--706 (1980; Zbl 0439.58014)], see also [\textit{R. H. Cushman} and \textit{L. M. Bates}, Global aspects of classical integrable systems. Basel: Birkhäuser (1997; Zbl 0882.58023)]. Hamiltonian monodromy refers to the monodromy of the torus bundle over a closed homotopically non-trivial path in the set of regular values of the energy-momentum map \((E,M_z)\) of the system. Non-trivial monodromy implies that the system has no global action variables. The lattice determined by the joint quantum spectrum was computed in [\textit{R. H. Cushman} and \textit{J. J. Duistermaat}, Bull. Am. Math. Soc., New Ser. 19, No. 2, 475--479 (1988; Zbl 0658.58039)]. It is defined as the set of values of the two action variables, one of them being \(M_z\) and the other denoted by \(I\), such that \(M_z = h n\), \(n \in \mathbb{Z}\) and \(I = h (m+1/2)\), \(m \in \mathbb{Z}_>\). In the same work it is also shown how the structure of the lattice implies the non-trivial monodromy of the spherical pendulum. This idea of understanding monodromy through the study of the level sets of action variables and the parallel transport of an elementary cell on that lattice was later significantly expanded by \textit{S. Vu Ngọc} in [Comm. Math. Phys. 203, No. 2, 465--479 (1999; Zbl 0981.35015)] and by \textit{B. I. Zhilinskií} and his collaborators [Acta Appl. Math. 87, No. 1-3, 281--307 (2005; Zbl 1073.37062); J. Phys. A: Math. Theor. 43, No. 43, 434033 (2010; Zbl 1202.81100)]. The present paper starts with an ad hoc study of the fibres of the energy-momentum map of the spherical pendulum. It gives a direct proof that fibres over regular values are two-dimensional tori while fibres over singular values are either periodic orbits or, in the two most exceptional cases, an isolated equilibrium or a pinched torus. Furthermore, it computes the bifurcation diagram (set of critical values of the energy-momentum map) for this system. Then the paper gives an explicit integral form for a second action variable \(I\) of the spherical pendulum (the first action being \(M_z\)). In particular, for \(M_z > 0\) it gives the expression \[ I(E,M_z) = \frac{1}{\pi} \int_{z_1}^{z_2} \frac{\sqrt{2(E-z)(1-z^2)-M_z^2}}{1-z^2}\, dz, \] where \(z_1 < z_2\) are roots of \(2(E-z)(1-z^2)-M_z^2\) in the interval \((-1,1)\). Then it extends the formula for \(I\) in a smooth way to \(M_z < 0\) separately for the cases \(E > 1\) and \(-1 < E < 1\), where \(E = 1\) is the energy of the singular value of the energy-momentum map. The need for different smooth extensions in the two cases is related to the monodromy and thus the non-existence of global smooth action variables. Based on these definitions the paper then numerically computes the integer action lattice. The integer action lattice is defined here as the set of points on the \(\{ M_z, E \}\)-plane where the actions \(\{ M_z, I \}\) are integer multiples of some arbitrary constant \(c > 0\). In other words, \(M_z = c n\), \(n \in \mathbb{Z}\) and \(I = c m\), \(m \in \mathbb{Z}_>\). In the physically oriented literature on monodromy, the constant \(c\) is usually being thought of as Planck's constant \(h\) entering through quantization and there is a minor modification in the desired level sets of \(I\) arising from non-zero Maslov indices, cf.\ the definitions in [\textit{R. H. Cushman} and \textit{J. J. Duistermaat}, loc. cit.]. Subsequently the paper proceeds to compute the monodromy of the spherical pendulum using the idea of parallel transport of an elementary cell in the numerically computed lattice, see also [Zbl 1073.37062]. The monodromy matrix is shown to be \[ \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) \] up to \(\mathrm{SL}(2,\mathbb{Z})\)-equivalence, as computed already in [\textit{J. J. Duistermaat}, Comm. Pure Appl. Math. 33, 687--706 (1980; Zbl 0439.58014)] for the spherical pendulum and predicted by the more general results for focus-focus singularities [\textit{V. S. Matveev}, Sb. Math. 187, No. 4, 495--524 (1996; Zbl 0871.58045)]; \textit{N. T. Zung} [Diff. Geom. Appl. 7, No. 2, 123--130 (1997; Zbl 0887.58023)]; \textit{S. Vu Ngọc} [Commun. Pure Appl. Math. 53, No. 2, 143--217 (2000; Zbl 1027.81012)]. Finally, the paper computes the topological type of the energy level sets using again the idea of parallel transport of an elementary cell along a path that starts at one boundary of the bifurcation diagram and ends at the other side. This computation gives the topological type of the corresponding lens space (or ``molecule of type \(A\)--\(A\)''). This proves the well-known fact [\textit{J. J. Duistermaat}, loc. cit.] that for \(E > 1\) the level set is an \(\mathbb{R}P^3\) while for \(-1 < E < 1\) it is an \(\mathbb{S}^3\).