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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, October 2015
1 
Tensor products of Dirac structures and interconnection in Lagrangian mechanics
Volume 6, Number 1, Pages: 67  98, 2014
Henry O. Jacobs
and Hiroaki Yoshimura
Abstract
References
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Many mechanical systems are large and complex, despite being composed of simple subsystems.
In order to understand such large systems it is natural to tear the system into these subsystems.
Conversely we must understand how to invert this tearing procedure.
In other words, we must understand interconnection of subsystems.
Such an understanding has been already shown in the context of Hamiltonian systems on vector spaces via
the portHamiltonian systems program, in which an interconnection may be achieved through the identification of shared variables, whereupon the notion of composition of Dirac structures allows one to interconnect two systems.
In this paper, we seek to extend the program of the portHamiltonian systems on vector spaces to the case of Lagrangian systems on manifolds and also extend the notion of composition of Dirac structures appropriately.
In particular, we will interconnect LagrangeDirac systems by modifying the respective Dirac structures of the involved subsystems.
We define the interconnection of Dirac structures via an interaction Dirac structure and a tensor product of Dirac structures.
We will show how the dynamics of the interconnected system is formulated as a function of the subsystems, and we will elucidate the associated variational principles.
We will then illustrate how this theory extends the theory of portHamiltonian systems and the notion of composition of Dirac structures to manifolds with couplings which do not require the identification of shared variables.
Lastly, we will show some examples: a massspring mechanical systems, an electric circuit, and a nonholonomic mechanical system.

2 
A HamiltonJacobi theory on Poisson manifolds
Volume 6, Number 1, Pages: 121  140, 2014
Manuel de León,
David Martín de Diego
and Miguel Vaquero
Abstract
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In this paper we develop a HamiltonJacobi theory in the setting of almost Poisson manifolds.
The theory extends the classical HamiltonJacobi theory and can be also applied to very general situations
including nonholonomic mechanical systems and time dependent systems with external forces.

3 
Andoyer's variables and phases in the free rigid body
Volume 6, Number 1, Pages: 25  37, 2014
Sebastián Ferrer
and Francisco J. Molero
Abstract
References
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Using Andoyer's variables we present a new proof of Montgomery's formula by measuring $\Delta\mu$ when $\nu$ has made a rotation. Our treatment is built on the equations of the differential system of the free rigid solid, together with the explicit expression of the spherical area defined by the intersection of the surfaces given by the energy and momentum integrals. We also consider the phase $\Delta\nu$ of the moving frame when $\mu$ has made a rotation around the angular momentum vector, and we give the formula for its computation.

4 
Aspects of reduction and transformation of Lagrangian systems with symmetry
Volume 6, Number 1, Pages: 1  23, 2014
E. GarcíaToraño Andrés,
Bavo Langerock
and Frans Cantrijn
Abstract
References
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This paper contains results on geometric Routh reduction and it is a continuation of a previous paper [7] where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry.

5 
Bundletheoretic methods for higherorder variational calculus
Volume 6, Number 1, Pages: 99  120, 2014
Michał Jóźwikowski
and Mikołaj Rotkiewicz
Abstract
References
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We present a geometric interpretation of the integrationbyparts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higherorder variational calculus.

6 
Fluidstructure interaction in the LagrangePoincaré formalism: The NavierStokes and inviscid regimes
Volume 6, Number 1, Pages: 39  66, 2014
Henry Jacobs
and Joris Vankerschaver
Abstract
References
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In this paper, we derive the equations of motion for an elastic body interacting with a perfect fluid via the framework of LagrangePoincaré reduction.
We model the combined fluidstructure system as a geodesic curve on the total space of a principal bundle on which a diffeomorphism group acts.
After reduction by the diffeomorphism group we obtain the fluidstructure interactions where the fluid evolves by the inviscid fluid equations.
Along the way, we describe various geometric structures appearing in fluidstructure interactions: principal connections, Lie groupoids, Lie algebroids, etc.
We finish by introducing viscosity in our framework as an external force and adding the noslip boundary condition.
The result is a description of an elastic body immersed in a NavierStokes fluid as an externally forced LagrangePoincaré equation.
Expressing fluidstructure interactions with LagrangePoincaré theory provides an alternative to the traditional description of the NavierStokes equations on an evolving domain.

7 
A note on the WehrheimWoodward category
Volume 3, Number 4, Pages: 507  515, 2012
Alan Weinstein
Abstract
References
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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.

8 
Linear
almost Poisson structures and HamiltonJacobi equation. Applications
to nonholonomic mechanics
Volume 2, Number 2, Pages: 159  198, 2010
Manuel de León,
Juan Carlos Marrero
and David Martín de Diego
Abstract
Full Text
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In this paper, we study the underlying geometry in the classical
HamiltonJacobi equation. The proposed formalism is also valid for
nonholonomic systems. We first introduce the essential geometric
ingredients: a vector bundle, a linear almost Poisson structure
and a Hamiltonian function, both on the dual bundle (a Hamiltonian
system). From them, it is possible to formulate the
HamiltonJacobi equation, obtaining as a particular case, the
classical theory. The main application in this paper is to
nonholonomic mechanical systems. For it, we first construct the
linear almost Poisson structure on the dual space of the vector
bundle of admissible directions, and then, apply the
HamiltonJacobi theorem. Another important fact in our paper is the use of the orbit theorem
to symplify the HamiltonJacobi equation, the introduction of the notion of morphisms preserving the
Hamiltonian system; indeed, this concept will be very useful to
treat with reduction procedures for systems with symmetries.
Several detailed examples are given to illustrate the utility of these new developments.

9 
LeibnizDirac structures and nonconservative systems with constraints
Volume 5, Number 2, Pages: 167  183, 2013
Ünver Çiftçi
Abstract
References
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Although conservative
Hamiltonian systems with
constraints can be formulated in
terms of Dirac structures,
a more general framework is
necessary to cover
also dissipative systems such as
gradient and metriplectic systems
with constraints.
We define LeibnizDirac structures which
lead to a natural generalization of Dirac
and Riemannian structures, for instance. From modeling point of
view, LeibnizDirac structures make it easy to formulate
implicit dissipative Hamiltonian systems.
We give their exact characterization in terms of
vector bundle maps from the tangent bundle to the
cotangent bundle and vice verse. Physical systems which can be formulated
in terms of LeibnizDirac structures are discussed.

10 
Higherorder mechanics: Variational principles and other topics
Volume 5, Number 4, Pages: 493  510, 2013
Pedro D. PrietoMartínez
and Narciso RománRoy
Abstract
References
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Related Articles
After reviewing the LagrangianHamiltonian unified formalism (i.e, the SkinnerRusk formalism)
for higherorder (nonautonomous) dynamical systems,
we state a unified geometrical version of the Variational Principles
which allows us to derive the
Lagrangian and Hamiltonian equations for these kinds of systems.
Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding
dynamical equations are recovered from this unified framework..

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