ISSN:
 1941-4889

eISSN:
 1941-4897

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Volume 11, 2019

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Journal of Geometric Mechanics

Open Access Articles

Particle relabelling symmetries and Noether's theorem for vertical slice models
Colin J. Cotter and Michael John Priestley Cullen
2019, 11(2): 139-151 doi: 10.3934/jgm.2019007 +[Abstract](905) +[HTML](276) +[PDF](394.81KB)
Abstract:

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the potential temperature. This is developed using the methodology of Cotter and Holm (Foundations of Computational Mathematics, 2012).

Preface
Martin Bauer, Martins Bruveris and Peter W. Michor
2017, 9(2): i-iii doi: 10.3934/jgm.201702i +[Abstract](918) +[HTML](673) +[PDF](31.6KB)
Abstract:
Preface
Pedro L. García and Manuel de León
2013, 5(4): v-vi doi: 10.3934/jgm.2013.5.4v +[Abstract](990) +[PDF](90.6KB)
Abstract:
This Special Issue of the Journal of Geometric Mechanics is devoted to the Proceedings of the Third Iberoamerican Meetings on Geometry, Mechanics and Control, which was held at the University of Salamanca (Spain) from September 3 until September 7, 2012.

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Preface
Juan-Pablo Ortega
2011, 3(4): ii-iv doi: 10.3934/jgm.2011.3.4ii +[Abstract](1145) +[PDF](209.6KB)
Abstract:
In July 2010 we celebrated the 60th birthday of Tudor S. Ratiu with a workshop entitled ``Geometry, Mechanics, and Dynamics" that was held at the Centre International de Rencontres Mathématiques in Luminy. Tudor is one of the world's most renowned and esteemed mathematicians and this conference was a great occasion to go over the numerous subjects on which he has worked and had a deep influence. Most of these topics are strongly connected to the subject matter of this journal, whose very foundation owes much to Tudor's encouragement and support.
    I coorganized this event with the late Jerry Marsden who was the main force behind it and started its preparation several years in advance as it was his habit with so many other things. Jerry was not only Tudor's PhD advisor but also his main collaborator for thirty years, friend, mentor and as for many of us, an endless source of intellectual inspiration. This is why we felt so sad when we learnt that his health had deteriorated and that he could not attend the meeting he had put so much dedication on and, needless to say, so devastated when the news of his passing arrived a few months later.

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A note on the Wehrheim-Woodward category
Alan Weinstein
2011, 3(4): 507-515 doi: 10.3934/jgm.2011.3.507 +[Abstract](1198) +[PDF](407.2KB)
Abstract:
Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.

2018  Impact Factor: 0.525

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