# American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

All Issues

## Journal of Geometric Mechanics

Open Access Articles

2020 doi: 10.3934/jgm.2020031 +[Abstract](193) +[HTML](93) +[PDF](432.95KB)
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A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.

2020, 12(3): ⅰ-ⅳ doi: 10.3934/jgm.2020028 +[Abstract](389) +[HTML](137) +[PDF](2018.06KB)
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2020, 12(1): ⅰ-ⅱ doi: 10.3934/jgm.2020001 +[Abstract](1253) +[HTML](281) +[PDF](235.95KB)
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2019, 11(4): ⅰ-ⅰ doi: 10.3934/jgm.2019033 +[Abstract](1134) +[HTML](137) +[PDF](125.95KB)
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2019, 11(2): ⅰ-ⅲ doi: 10.3934/jgm.2019016 +[Abstract](1347) +[HTML](292) +[PDF](5612.9KB)
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2019, 11(2): 139-151 doi: 10.3934/jgm.2019007 +[Abstract](3701) +[HTML](590) +[PDF](394.81KB)
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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).

2017, 9(2): i-iii doi: 10.3934/jgm.201702i +[Abstract](1488) +[HTML](1047) +[PDF](31.6KB)
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2013, 5(4): v-vi doi: 10.3934/jgm.2013.5.4v +[Abstract](1637) +[PDF](90.6KB)
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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.

2013, 5(4): i-iii doi: 10.3934/jgm.2013.5.4i +[Abstract](1877) +[PDF](110.3KB)
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We would like to render our tribute to Professor Pedro Luis García , who gave a great contribution to the development of Differential Geometry and Mathematical Physics in Spain.

2012, 4(3): i-i doi: 10.3934/jgm.2012.4.3i +[Abstract](1532) +[PDF](89.8KB)
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This number is the third one that the Journal of Geometric Mechanics dedicates to the works presented on the occasion of the 60th birthday celebration of Tudor S. Ratiu, held at the Centre International de Rencontres Mathématiques de Luminy. As it was the case for the first two numbers (volume 3(4) and volume (4)2), we hope that the articles contained in this special number will give the reader an appreciation for the importance of the contributions of Tudor S. Ratiu to the fields that are at the core of this journal.
2012, 4(2): i-i doi: 10.3934/jgm.2012.4.2i +[Abstract](1667) +[PDF](89.0KB)
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This number is the second one that the Journal of Geometric Mechanics dedicates to the works presented on the occasion of the 60th birthday celebration of Tudor S. Ratiu, held at the Centre International de Rencontres Math\'ematiques. As it was the case for the first number (volume 3(4)), we hope that the articles contained in this special number will give the reader an appreciation for the importance of the contributions of Tudor S. Ratiu to the fields that are at the core of this journal.
2011, 3(4): ii-iv doi: 10.3934/jgm.2011.3.4ii +[Abstract](1679) +[PDF](209.6KB)
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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.

2011, 3(4): 507-515 doi: 10.3934/jgm.2011.3.507 +[Abstract](1905) +[PDF](407.2KB)
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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.
2009, 1(3): i-i doi: 10.3934/jgm.2009.1.3i +[Abstract](1739) +[PDF](27.2KB)
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The Hamilton-Jacobi theory is a classical subject that was extensively developed in the last two centuries. The Hamilton-Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. Indeed, the powerful of this method is that, in spite of the difficulties to solve a partial differential equation instead of an ordinary differential one, in many cases it works, being an extremely useful tool, usually more than Hamilton’s equations. Indeed, in these cases the method provides an immediate way to integrate the equations of motion. The modern interpretation relating the Hamilton-Jacobi procedure with the theory of lagrangian submanifolds is an important source of new results and insights.
In addition, the Hamilton-Jacobi-Bellman equation is a partial differential equation which is central to optimal control theory. The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton-Jacobi equation by Hamilton and Jacobi.
This special issue on Hamilton-Jacobi theory wants to bring specialists coming from different areas of research and show how the Hamilton-Jacobi theory is so useful in their domains: completely integrable systems, nonholonomic mechanics, Schrödinger equation, optimal control theory, and, in particular, applications in engineering and economics.

2019  Impact Factor: 0.649