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Xiudi Tang

Publications and source records attributed to Xiudi Tang.

9 recordsLinked to original sources
arXiv2025

Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

We study the spherical pendulum system with an arbitrary potential function $V = V (z)$, which is an integrable system with a first integral whose Hamiltonian flow is periodic. We give an explicit solution to this integrable system and then we compute its action-angle coordinates. In the special case where the potential function is symmetric quadratic like $V = z^2$, we represent its action-angle coordinates in terms of elliptic integrals, and calculate the monodromy.

math.SG↗
arXiv2024

A note on the symplectic classification of almost-toric systems

Since simple semitoric systems were classified about fifteen years ago, and semitoric systems five years ago, we want to move a step forward to almost-toric systems. We give a classification of compact almost-toric systems in dimension four up to fiber-preserving symplectomorphisms, in terms of the base, Taylor series, and twisting indices, analogous to the five invariants for semitoric systems. For convenience, we specify an ordering of focus-focus values and a choice of two cut rays at each of them.

math.SG↗
arXiv2024

Classifying affine structures with focus-focus singularities

We study the singular affine structures of integrable systems with focus-focus singular fibers on the image of momentum maps. The classification of singular affine structures is equivalent to the classification of simple semitoric systems up to fiber-preserving symplectomorphisms but is not equivalent for semitoric systems with multiple pinched fibers and we give counterexamples for any number of more than one pinch on each fiber.

math.SG↗
arXiv2021

Symplectic excision

We use time-independent incomplete Hamiltonian flows to excise interesting closed subsets of positive codimension from symplectic manifolds. Examples of such subsets include what we call a "Cantor brush", a "box with a tail", and -- more generally -- epigraphs of lower semicontinuous functions. This answers a question of Alan Weinstein about excision of a ray, and it generalizes a result of Bernd Stratmann about excision of the product of a ray with a manifold.

math.SG↗
arXiv2019

Semitoric systems of non-simple type

Within integrable systems, the class of so called "semitoric" integrable systems in dimension four has attracted a lot of attention in recent years, especially since fundamental examples from classical and quantum mechanics have been identified as semitoric by different groups of researchers. Several of these examples, however, show a particular trait not included in the original theory, that is, the presence of multiple (i.e. two or more) rank zero isolated singularities in the same energy-momentum level sets. Systems with this property are called non-simple. This paper extends the original theory of Pelayo and V\~u Ngoc to non-simple systems.

math.SG↗
arXiv2019

Removing a ray from a noncompact symplectic manifold

We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism in the case of the standard Euclidean space. We use this excision trick to construct a nowhere vanishing Liouville vector fields on every cotangent bundle.

math.SG↗
arXiv2018

Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing $k > 1$ focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(\mathbb{Z}_2 \times D_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold $(M,\omega)$ near the singular fiber containing the $k$ focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.

math.SG↗
arXiv2018

Symplectic stability on manifolds with cylindrical ends

A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural restricted setting for the study of analogs of Moser's symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak-Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and completeness assumptions on the symplectic forms at infinity we show that symplectic stability holds under a natural growth condition on the path of symplectic forms. The result can be straightforwardly applied as we show through explicit examples.

math.SG↗