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September 2018, 10(3): 331-357. doi: 10.3934/jgm.2018012
A family of compact semitoric systems with two focus-focus singularities
Sonja Hohloch 1,, and Joseph Palmer 2,
University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, B-2020 Antwerpen, Belgium
Rutgers University, Department of Mathematics, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
* Corresponding author: Sonja Hohloch
Received October 2017 Revised June 2018 Published August 2018
Fund Project: The first author was partially supported by the Research Fund of the University of Antwerp and the second author is partially supported by an AMS-Simons travel grant.
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About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngọc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a $6$-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.
Keywords: Symplectic geometry, integrable system, semitoric integrable system, Hamiltonian system, momentum map, focus-focus singularity.
Mathematics Subject Classification: Primary: 37J35; Secondary: 53D20.
Citation: Sonja Hohloch, Joseph Palmer. A family of compact semitoric systems with two focus-focus singularities. Journal of Geometric Mechanics, 2018, 10 (3) : 331-357. doi: 10.3934/jgm.2018012
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Figure 1. An image of the momentum map $(J_{(1, 2)}, H_{(s_1, s_2)})$ with the rank 0 points marked for varying values of $s_1, s_2\in [0, 1]$. Notice that the coupled angular momenta system shown in Figure 3 is the bottom row of the system shown in this figure since the coupled angular momenta is the special case for which $s_2 = 0$
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Figure 2. Left: a plot of the set $\gamma$, which is the union of $\gamma_{(S, N)}$ (blue) and $\gamma_{(N, S)}$ (orange), see Equation (23). Right: Values of $(s_1, s_2)$ for which the system $(J_{(1, 2)}, H_{(s_1, s_2)})$ has focus-focus values at: only the point $(S, N)$ (blue), only the point $(N, S)$ (orange), or at both points (green). The system is degenerate on the black curves. Compare with Figure 1
Figure 4. Four semitoric polygons associated to the system (1). The slanted edges all have slope $\pm 1$. For each polygon the $x$-coordinates of the vertices, from left to right, are $-R_1-R_2$, $R_1-R_2$, $-R_1+R_2$, and $R_1+R_2$ (since we assume $R_1<R_2$)
Figure 3. The momentum map image for the coupled angular momenta system with the rank zero points marked. As the coupling parameter $t$ changes one of the rank zero points transitions from being elliptic-elliptic to being focus-focus and then back to elliptic-elliptic
Figure 5. This figure analyses the right hand side of Equation (22): The plot on the left shows the graph of the right hand side of Equation (22) which is always below $-1.06066$. The contour plot on the right displays the associated level sets
Figure 6. Case $(e_1, e_2) = (1, 1)$: on the left, the graph of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, 1))}$ (orange) and a plane through zero (blue) are displayed. On the right, the associated level sets of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, 1))}$ are shown
Figure 7. Case $(e_1, e_2) = (1, -1)$: on the left, the graph of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, -1))}$ (orange) and a plane through zero (blue) are displayed. On the right, the associated level sets of $ (s_1, s_2) \mapsto \Delta _{((s_1, s_2), (1, -1))}$ are shown
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Sonja Hohloch Joseph Palmer