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Journal of Geometric Mechanics

2011 , Volume 3 , Issue 4

Special issue dedicated to Tudor S. Ratiu on the occasion of his 60th birthday
Guest Editor: Juan-Pablo Ortega

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Preface
Juan-Pablo Ortega
2011, 3(4): ii-iv doi: 10.3934/jgm.2011.3.4ii +[Abstract](540) +[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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Un-reduction
Martins Bruveris, David C. P. Ellis, Darryl D. Holm and François Gay-Balmaz
2011, 3(4): 363-387 doi: 10.3934/jgm.2011.3.363 +[Abstract](494) +[PDF](471.8KB)
Abstract:
This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally conceived for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.
Sobolev metrics on shape space of surfaces
Martin Bauer, Philipp Harms and Peter W. Michor
2011, 3(4): 389-438 doi: 10.3934/jgm.2011.3.389 +[Abstract](703) +[PDF](2091.6KB)
Abstract:
Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \overline{g}( P^fh, k) vol (f^*\overline{g})$$ where $\overline{g}$ is some fixed metric on $N$, $f^*\overline{g}$ is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\overline{g}$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.
Point vortices on the sphere: Stability of symmetric relative equilibria
Frederic Laurent-Polz, James Montaldi and Mark Roberts
2011, 3(4): 439-486 doi: 10.3934/jgm.2011.3.439 +[Abstract](608) +[PDF](2017.1KB)
Abstract:
We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.
Covariantizing classical field theories
Marco Castrillón López and Mark J. Gotay
2011, 3(4): 487-506 doi: 10.3934/jgm.2011.3.487 +[Abstract](689) +[PDF](456.7KB)
Abstract:
We show how to enlarge the covariance groups of a wide variety of classical field theories in such a way that the resulting ``covariantized'' theories are `essentially equivalent' to the originals. In particular, our technique will render many classical field theories generally covariant, that is, the covariantized theories will be spacetime diffeomorphism-covariant and free of absolute objects. Our results thus generalize the well-known parametrization technique of Dirac and Kuchař. Our constructions apply equally well to internal covariance groups, in which context they produce natural derivations of both the Utiyama minimal coupling and Stückelberg tricks.
A note on the Wehrheim-Woodward category
Alan Weinstein
2011, 3(4): 507-515 doi: 10.3934/jgm.2011.3.507 +[Abstract](499) +[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.

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