2023/02/26 by Diana M. H. Nguyen, Sean R. Dawson, Nguyen, Diana M. H. +3
Environmental Science · Mathematics · Physics and Astronomy · #37J15 #37J35 #53D20 #53D22 #81Q20 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Methane Hydrates and Related Phenomena #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2302.13270
openalex publication_date 2023/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the space of orthogonally separable coordinates on the sphere S3 induces a natural family of integrable systems, which after symplectic reduction leads to a family of integrable systems on S2 × S2. The generic member of the family corresponds to ellipsoidal coordinates. We use the theory of compatible Poisson structures to study the critical points and critical values of the momentum map. Interesting structure arises because the ellipsoidal coordinate system can degenerate in a variety of ways, and all possible orthogonally separable coordinate systems on S3 (including degenerations) have the topology of the Stasheff polytope K4, which is a pentagon. We describe how the generic integrable system degenerates, and how the appearance of global SO(2) and SO(3) symmetries is the main feature that organises the various degenerate systems. For the whole family we show that there is an action map whose image is an equilateral triangle. When higher symmetry is present, this triangle ``unfolds'' into a semi-toric polygon (when there is one global S1-action) or a Delzant polygon (when there are two global S1-actions). We believe that this family of integrable systems is a natural playground for theories of global symplectic classification of integrable systems.