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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 2014 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, August 2014
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 
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
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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.

3 
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
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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.

4 
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.

5 
Bundletheoretic methods for higherorder variational calculus
Volume 6, Number 1, Pages: 99  120, 2014
Michał Jóźwikowski
and Mikołaj Rotkiewicz
Abstract
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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
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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
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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 
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
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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.

9 
The Toda lattice, old and new
Volume 5, Number 4, Pages: 511  530, 2013
Carlos Tomei
Abstract
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Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also nonJacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra.
The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.

10 
On Euler's equation and 'EPDiff'
Volume 5, Number 3, Pages: 319  344, 2013
David Mumford
and Peter W. Michor
Abstract
References
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We study a family of approximations to Euler's equation depending on two parameters
$\epsilon,η \ge 0$. When $\epsilon = η = 0$ we have Euler's equation and when both are positive we have
instances of the class of integrodifferential equations called EPDiff in imaging science. These
are all geodesic equations on either the full diffeomorphism group
${Diff}_{H^\infty}(\mathbb{R}^n)$ or, if $\epsilon = 0$, its volume preserving subgroup. They are
defined by the right invariant metric induced by the norm on vector fields given by
$$ v_{\epsilon,η} = \int_{\mathbb{R}^n} \langle L_{\epsilon,η} v, v \rangle\, dx $$
where $L_{\epsilon,η} = (I\frac{η^2}{p} \triangle)^p \circ (I\frac {1}{\epsilon^2} \nabla \circ div)$.
All geodesic equations are locally wellposed, and the $L_{\epsilon,η}$equation admits solutions for
all time if $η > 0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by
estimates which, however, are only local in time. This approach leads to a new notion of momentum
which is transported by the flow and serves as a generalization of vorticity. We also discuss how
delta distribution momenta lead to ``vortexsolitons", also called ``landmarks" in imaging science,
and to new numeric approximations to fluids.

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