Examples for obstructions to action-angle coordinates

1988 ◽  
Vol 110 (1-2) ◽  
pp. 27-30 ◽  
Author(s):  
Larry M. Bates
Keyword(s):  
Global Existence ◽  
Symplectic Manifolds ◽  
Torus Bundles

SynopsisWe give examples of symplectic manifolds which are also non-trivial principal torus-bundles with Lagrangian fibres. These bundles are examples of spaces with an obstruction to the global existence of action-angle variables.

scholarly journals Integrable Systems on Singular Symplectic Manifolds: From Local to Global

2021 ◽  
Author(s):  
Robert Cardona ◽  
Eva Miranda
Keyword(s):  
Global Existence ◽  
Integrable System ◽  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Form ◽  
Symplectic Manifolds ◽  
Symplectic Structures ◽  
Critical Set ◽  
Symplectic Forms ◽  
Topological Obstructions

Abstract In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a $b$-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [34] and [35] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and $b$-symplectic forms in [34]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set $Z$ of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on $Z$.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

The Affective Politics of Karma among Buddhist Cancer Charities in Vietnam

2020 ◽  
Vol 15 (4) ◽  
pp. 33-62
Author(s):  
Sara Swenson
Keyword(s):  
Public Health ◽  
Global Existence ◽  
Food Production ◽  
Public Health Intervention ◽  
Health Intervention ◽  
Moral Failure ◽  
Cancer Rates ◽  
Charity Work

In this article, I explore how Buddhist charity workers in Vietnam interpret rising cancer rates through understandings of karma. Rather than framing cancer as a primarily physical or medical phenomenon, volunteers state that cancer is a product of collective moral failure. Corruption in public food production is both caused by and perpetuates bad karma, which negatively impacts global existence. Conversely, charity work creates merit, which can improve collective karma and benefit all living beings. I argue that through such interpretations of karma, Buddhist volunteers understand their charity at cancer hospitals as an affective and ethical form of public health intervention.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals Global existence of null-form wave equations in exterior domains

2007 ◽  
Vol 256 (3) ◽  
pp. 521-549 ◽  
Author(s):  
Jason Metcalfe ◽  
Christopher D. Sogge
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Exterior Domains
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