Abstract
We construct a one-parameter family F_t=(J, H_t)_{0 le t le 1} of integrable systems on a compact 4-dimensional symplectic manifold (M, omega ) that changes smoothly from a toric system F_0 with eight elliptic–elliptic singular points via toric type systems to a semitoric system F_t for t^-< t < t^+. These semitoric systems F_t have precisely four elliptic–elliptic and four focus–focus singular points. Moreover, at t= frac{1}{2}, the system has precisely two focus–focus fibres each of which contains exactly two focus–focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.
Highlights
Integrable systems lie at the intersection of many areas in mathematics and physics like, for example, dynamical systems, ODEs, PDEs, symplectic geometry, Lie theory, classical mechanics, mathematical physics, etc
Integrable systems display a ‘certain amount of order’ due to being ‘not chaotic’ and having their flow lines stay in the fibres
Since a completely integrable system gives rise to a Lagrangian fibration this explicit family may become of interest for the study of low dimensional singular Lagrangian fibrations and, for instance, thedisplaceability of its fibers under Hamiltonian isotopies, i.e. questions of symplectic rigidity
Summary
Integrable systems lie at the intersection of many areas in mathematics and physics like, for example, dynamical systems, ODEs, PDEs, symplectic geometry, Lie theory, classical mechanics, mathematical physics, etc.
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