JGM
Foreword
Juan-Pablo Ortega
Journal of Geometric Mechanics 2012, 4(2): i-i doi: 10.3934/jgm.2012.4.2i
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.
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JGM
The stochastic Hamilton-Jacobi equation
Joan-Andreu Lázaro-Camí Juan-Pablo Ortega
Journal of Geometric Mechanics 2009, 1(3): 295-315 doi: 10.3934/jgm.2009.1.295
We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system.
keywords: Hamilton-Jacobi equation. Hamiltonian stochastic differential equation stochastic differential equation
JGM
Preface
Juan-Pablo Ortega
Journal of Geometric Mechanics 2011, 3(4): ii-iv doi: 10.3934/jgm.2011.3.4ii
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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JGM
Foreword
Juan-Pablo Ortega
Journal of Geometric Mechanics 2012, 4(3): i-i doi: 10.3934/jgm.2012.4.3i
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.
keywords:
JGM
Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies
Lyudmila Grigoryeva Juan-Pablo Ortega Stanislav S. Zub
Journal of Geometric Mechanics 2014, 6(3): 373-415 doi: 10.3934/jgm.2014.6.373
This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the ``orbitron". We study the nonlinear stability of a branch of equatorial relative equilibria using the energy-momentum method and we provide sufficient conditions for their $\mathbb{T}^2$--stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that allow us to conclude the sharpness of some of the nonlinear stability conditions obtained.
keywords: Hamiltonian systems with symmetry relative equilibrium nonlinear stability/instability. momentum maps orbitron magnetic systems generalized orbitron

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