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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. Infinitedimensional 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 2015 in March, June, September and December and is a publication of the American Institute of Mathematical Sciences. All rights reserved.
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TOP 10 Most Read Articles in JGM, April 2015
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 nonstandard 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 solidfluid 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
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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
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A discrete Lagrange problem is defined as a discrete Lagrangian system endowed with a constraint submanifold in the space of 1jets 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 1form, 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
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The procedure of linearizing a controlaffine system along a nontrivial
reference trajectory is studied from a differential geometric perspective. A
coordinateinvariant 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 timevarying 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 
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 3manifold $S^2 \times S^1$.
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.

6 
Variational principles for spin systems and the Kirchhoff rod
Volume 1, Number 4, Pages: 417  444, 2010
François GayBalma,
Darryl D. Holm
and Tudor S. Ratiu
Abstract
Full Text
Related Articles
We obtain the affine EulerPoincaré equations by standard Lagrangian reduction and deduce the associated Clebschconstrained 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.

7 
Semibasic 1forms 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 FrölicherNijenhuis theory to obtain global Helmholtz
conditions, expressed in terms of a semibasic 1form, 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 semibasic 1form is 1homogeneous (0homogeneous) 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.

8 
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
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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.

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
Leibniztype property. The bilinear bracket is not, in general,
skewsymmetric and it does not satisfy, in general, the Jacobi
identity. Since skewsymmetry 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 GarcíaNaranjo
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
$\SO(n1)$. We show how this symmetry group can be factored out, and
pass to the fully reduced system on (a fiber bundle over)
T^{*}$S^{n1}$.
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^{*}$S^{n1}$level. Moreover, the
$3$dimensional
ball becomes Hamiltonian at the T^{*}$S^{2}$level after
time reparametrization, whereby we reprove a result of
[4, 5] in symplectogeometric terms.
We also study compression followed by reduction of generalized
Chaplygin systems.

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