Integer lattices of action-angle variables for ``spherical pendulum'' system
type: Item
instance of: scholarly article
title: Integer lattices of action-angle variables for ``spherical pendulum'' system
zbMATH Open document ID: 1315.37033
DOI: 10.3103/S0027132214040019
author: Elena O. Kantonistova
published in: Moscow University Mathematics Bulletin
publication date: 2015-03-24T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 53D20
zbMATH DE Number: 6418917
zbMATH Keywords: integrable Hamiltonian systems
zbMATH Keywords: Hamiltonian monodromy
zbMATH Keywords: action-angle variables
reviewed by: Konstantinos Efstathiou
MaRDI profile type: Publication
cites work: Q4939713
cites work: Q3761318
cites work: Q3361570
cites work: Q2785808
cites work: The quantum mechanical spherical pendulum
cites work: Q4483003
cites work: Integer lattices of the action variables for the generalized Lagrange case
full work available at URL: s0027132214040019
OpenAlex ID: W2011491966
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description: scientific article; zbMATH DE number 6418917