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The Journal of Geometric Mechanics (JGM)

ISSN 1941-4889(print) ISSN 1941-4897(online)	Current volume Go	Journal archive Go	Advanced Search
	The Journal of Geometric Mechanics (JGM) aims to publish research articles devoted to geometric methods (in a broad sense) in mechanics and control theory, and intends to facilitate interaction between theory and applications. Advances in the following topics are welcomed by the journal: 1. Lagrangian and Hamiltonian mechanics 2. Symplectic and Poisson geometry and their applications to mechanics 3. Geometric and optimal control theory 4. Geometric and variational integration 5. Geometry of stochastic systems 6. Geometric methods in dynamical systems 7. Continuum mechanics 8. Classical field theory 9. Fluid mechanics 10. Infinite-dimensional dynamical systems 11. Quantum mechanics and quantum information theory 12. Applications in physics, technology, engineering and the biological sciences More detailed information on the subjects covered by the journal can be found by viewing the fields of research of the members of the editorial board. The journal is published by the American Institute of Mathematical Sciences, with the support of the Consejo Superior de Investigaciones Científicas (CSIC). Contributions to this journal are published free of charge. JGM will have four issues published in 2013 in March, June, September and December and is a publication of the American Institute of Mathematical Sciences. All rights reserved.

TOP 10 Most Read Articles in JGM, May 2012

1	The geometry and dynamics of interacting rigid bodies and point vortices Volume 1, Number 2, Pages: 223 - 266, 2009 Joris Vankerschaver, Eva Kanso and Jerrold E. Marsden Abstract Full Text Related Articles We derive the equations of motion for a planar rigid body of circular shape moving in a 2D perfect fluid with point vortices using symplectic reduction by stages. After formulating the theory as a mechanical system on a configuration space which is the product of a space of embeddings and the special Euclidian group in two dimensions, we divide out by the particle relabeling symmetry and then by the residual rotational and translational symmetry. The result of the first stage reduction is that the system is described by a non-standard magnetic symplectic form encoding the effects of the fluid, while at the second stage, a careful analysis of the momentum map shows the existence of two equivalent Poisson structures for this problem. For the solid-fluid system, we hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden, Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the other hand. As a side result, we obtain a convenient expression for the symplectic leaves of the reduced system and we shed further light on the interplay between curvatures and cocycles in the description of the dynamics.
2	When is a control system mechanical? Volume 2, Number 3, Pages: 265 - 302, 2010 Sandra Ricardo and Witold Respondek Abstract References Full Text Related Articles In this work we present a geometric setting for studying mechanical control systems. We distinguish a special class: the class of geodesically accessible mechanical systems, for which the uniqueness of the mechanical structure is guaranteed (up to an extended point transformation). We characterise nonlinear control systems that are state equivalent to a system from this class and we describe the canonical mechanical structure attached to them. Several illustrative examples are given.
3	Variational integrators for discrete Lagrange problems Volume 2, Number 4, Pages: 343 - 374, 2011 Pedro L. García, Antonio Fernández and César Rodrigo Abstract References Full Text Related Articles A discrete Lagrange problem is defined as a discrete Lagrangian system endowed with a constraint submanifold in the space of 1-jets of the discrete fibred manifold that configures the system. After defining the concepts of admissible section and infinitesimal admissible variation, the objective of these problems is to find admissible sections that are critical for the Lagrangian of the system with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). Next, we define the concept of Cartan 1-form, establish a Noether theory for symmetries and introduce a notion of "constrained variational integrator" that we characterize through a Cartan equation ensuring its symplecticity. Under a certain regularity condition of the problem of Lagrange, we prove the existence and uniqueness of this kind of integrators in the neighborhood of a critical section, showing then that such integrators can be constructed from a generating function of the second class in the sense of symplectic geometry. Finally, the whole theory is illustrated with three elementary examples.
4	Geometric Jacobian linearization and LQR theory Volume 2, Number 4, Pages: 397 - 440, 2011 Andrew D. Lewis and David R. Tyner Abstract References Full Text Related Articles The procedure of linearizing a control-affine system along a non-trivial reference trajectory is studied from a differential geometric perspective. A coordinate-invariant setting for linearization is presented. With the linearization in hand, the controllability of the geometric linearization is characterized using an alternative version of the usual controllability test for time-varying linear systems. The various types of stability are defined using a metric on the fibers along the reference trajectory and Lyapunov's second method is recast for linear vector fields on tangent bundles. With the necessary background stated in a geometric framework, linear quadratic regulator theory is understood from the perspective of the Maximum Principle. Finally, the resulting feedback from solving the infinite time optimal control problem is shown to uniformly asymptotically stabilize the linearization using Lyapunov's second method.
5	An elementary derivation of the Montgomery phase formula for the Euler top Volume 2, Number 1, Pages: 113 - 118, 2010 José Natário Abstract Full Text Related Articles We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the $2$-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.
6	Impulsive control of a symmetric ball rolling without sliding or spinning Volume 2, Number 4, Pages: 321 - 342, 2011 Hernán Cendra, María Etchechoury and Sebastián J. Ferraro Abstract References Full Text Related Articles A ball having two of its three moments of inertia equal and whose center of mass coincides with its geometric center is called a symmetric ball. The free dynamics of a symmetric ball rolling without sliding or spinning on a horizontal plate has been studied in detail in a previous work by two of the authors, where it was shown that the equations of motion are equivalent to an ODE on the 3-manifold $S^2 \times S^1$. In this paper we present an approach to the impulsive control of the position and orientation of the ball and study the speed of convergence of the algorithm. As an example we apply this approach to the solutions of the isoparallel problem.
7	Variational principles for spin systems and the Kirchhoff rod Volume 1, Number 4, Pages: 417 - 444, 2010 François Gay-Balma, Darryl D. Holm and Tudor S. Ratiu Abstract Full Text Related Articles We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.
8	Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations Volume 1, Number 2, Pages: 159 - 180, 2009 Ioan Bucataru and Matias F. Dahl Abstract Full Text Related Articles We use Frölicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is a Lagrangian vector field. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.
9	The ubiquity of the symplectic Hamiltonian equations in mechanics Volume 1, Number 1, Pages: 1 - 34, 2009 P. Balseiro, M. de León, Juan Carlos Marrero and D. Martín de Diego Abstract Full Text Related Articles In this paper, we derive a "Hamiltonian formalism" for a wide class of mechanical systems, that includes, as particular cases, classical Hamiltonian systems, nonholonomic systems, some classes of servomechanisms... This construction strongly relies on the geometry characterizing the different systems. The main result of this paper is to show how the general construction of the Hamiltonian symplectic formalism in classical mechanics remains essentially unchanged starting from the more general framework of algebroids. Algebroids are, roughly speaking, vector bundles equipped with a bilinear bracket of sections and two vector bundle morphisms (the anchors maps) satisfying a Leibniz-type property. The bilinear bracket is not, in general, skew-symmetric and it does not satisfy, in general, the Jacobi identity. Since skew-symmetry is related with preservation of the Hamiltonian, our Hamiltonian framework also covers some examples of dissipative systems. On the other hand, since the Jacobi identity is related with the preservation of the associated linear Poisson structure, then our formalism also admits a Hamiltonian description for systems which do not preserve this Poisson structure, like nonholonomic systems. Some examples of interest are considered: gradient extension of dynamical systems, nonholonomic mechanics and generalized nonholonomic mechanics, showing the applicability of our theory and constructing the corresponding Hamiltonian formalism.
10	$G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball Volume 1, Number 1, Pages: 35 - 53, 2009 Simon Hochgerner and Luis García-Naranjo Abstract Full Text Related Articles Via compression ([18, 8]) we write the $n$-dimensional Chaplygin sphere system as an almost Hamiltonian system on T^$\SO(n)$ with internal symmetry group $\SO(n-1)$. We show how this symmetry group can be factored out, and pass to the fully reduced system on (a fiber bundle over) T^$S^{n-1}$. This approach yields an explicit description of the reduced system in terms of the geometric data involved. Due to this description we can study Hamiltonizability of the system. It turns out that the homogeneous Chaplygin ball, which is not Hamiltonian at the T^$\SO(n)$-level, is Hamiltonian at the T^$S^{n-1}$-level. Moreover, the $3$-dimensional ball becomes Hamiltonian at the T^*$S^{2}$-level after time reparametrization, whereby we re-prove a result of [4, 5] in symplecto-geometric terms. We also study compression followed by reduction of generalized Chaplygin systems.

2011 Impact Factor.812