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تسجيل مستخدم جديدSemitoric systems of non-simple type
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A semitoric integrable system $F=(J,H)$ on a symplectic $4$-manifold is simple if each fiber of $J$ contains at most one focus-focus critical point. Simple semitoric systems were classified about ten years ago by Pelayo-V~u Ngoc in terms of five invariants. In this paper we explain how the simplicity assumption can be removed from the classification by adapting the invariants.
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About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc 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.
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