Taylor series and twisting-index invariants of coupled spin-oscillators

Abstract

About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vu Ngoc by means of five invariants. A standard example of such a system is the coupled spin-oscillator on ${\mathbb{S}}^2\times {\mathbb{R}}^2$. Calculations of three of the five semitoric invariants of this system (namely the number of focus-focus singularities, the generalised semitoric polygon, and the height invariant) already appeared in the literature, but the so-called twisting index was not yet computed and, of the so-called Taylor series invariant, only the linear terms were known. In the present paper, we complete the list of invariants for the coupled spin-oscillator by calculating higher order terms of the Taylor series invariant and by computing the twisting index. Moreover, we prove that the Taylor series invariant has certain symmetry properties that make the even powers in one of the variables vanish and allow us to show superintegrability of the coupled spin-oscillator on the zero energy level.

38 pages, 6 figures. Minor details of the exposition made clearer and small sign mistake in intermediate steps fixed

References in corpus (2)

• Vanishing Twist near Focus-Focus Points
• Vanishing Twist in the Hamiltonian Hopf Bifurcation

Cited by in corpus (7)

• Symplectic classification of coupled angular momenta
• The height invariant of a four-parameter semitoric system with two focus-focus singularities
• Constructions of b-semitoric systems
• Survey on recent developments in semitoric systems
• Packing Densities of Delzant and Semitoric Polygons
• The $p$-adic Jaynes-Cummings model in symplectic geometry
• The twisting index in semitoric systems