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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 2016 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 2016
1 
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.

2 
A unifying mechanical equation with applications to nonholonomic constraints and dissipative phenomena
Volume 7, Number 4, Pages: 473  482, 2015
E. Minguzzi
Abstract
References
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A mechanical covariant equation is introduced which retains all the effectingness of the Lagrange equation while being able to describe, in a unified way, other phenomena including friction, nonholonomic constraints and energy radiation (LorentzAbrahamDirac force equation).
A quantization rule adapted to the dissipative degrees of freedom is proposed which does not pass through the variational formulation.

3 
Symmetry reduction, integrability and reconstruction in $k$symplectic field theory
Volume 7, Number 4, Pages: 395  429, 2015
L. Búa,
T. Mestdag
and M. Salgado
Abstract
References
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We investigate the reduction process of a $k$symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the socalled LagrangePoincaré field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a $k$connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' $k$connection.

4 
Computing metamorphoses
between discrete measures
Volume 5, Number 1, Pages: 131  150, 2013
Casey L. Richardson
and Laurent Younes
Abstract
References
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Metamorphosis is a mathematical framework for diffeomorphic pattern matching in which one defines a distance
on a space of images or shapes. In the case of image matching, this distance involves computing the energetically optimal
way in which one image can be morphed into the other, combining both smooth deformations and changes in the image intensity.
In [12], Holm, Trouvé and Younes studied the metamorphosis of more singular deformable objects, in particular measures.
In this paper, we present results on the analysis and computation of discrete measure metamorphosis, building upon the work in [12].
We show that, when matching sums of Dirac measures, minimizing evolutions can include other singular distributions, which complicates the numerical approximation of such solutions.
We then present an Eulerian numerical scheme that accounts for these distributions, as well as some numerical experiments using this scheme.

5 
Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures
Volume 7, Number 4, Pages: 483  515, 2015
Giovanni Rastelli
and Manuele Santoprete
Abstract
References
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We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Using this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinateindependent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on $ \mathfrak{ so}^\ast (3) $ and $ \mathfrak{ so}^\ast (4)$) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

6 
Geometric arbitrage theory and market dynamics
Volume 7, Number 4, Pages: 431  471, 2015
Simone Farinelli
Abstract
References
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We have embedded the classical theory of stochastic finance into a
differential geometric framework called Geometric Arbitrage
Theory and show that it is possible to:
$\bullet$ Write arbitrage as curvature of a principal fibre bundle.
$\bullet$ Parameterize arbitrage strategies by its holonomy.
$\bullet$ Give the Fundamental Theorem of Asset Pricing a
differential homotopic characterization.
$\bullet$ Characterize Geometric Arbitrage Theory by five principles and
show they are consistent with the classical theory of
stochastic finance.
$\bullet$ Derive for a closed market the equilibrium solution for market portfolio and
dynamics in the cases where:
 Arbitrage is allowed but minimized.
 Arbitrage is not allowed.
$\bullet$ Prove that the nofreelunchwithvanishingrisk condition
implies the zero curvature condition. The converse is in general
not true and additionally requires the Novikov condition for the
instantaneous Sharpe Ratio to be satisfied.

7 
The Toda lattice, old and new
Volume 5, Number 4, Pages: 511  530, 2013
Carlos Tomei
Abstract
References
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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.

8 
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
References
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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.

9 
Invariant metrics on Lie groups
Volume 7, Number 4, Pages: 517  526, 2015
Gerard Thompson
Abstract
References
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Index formulas for the curvature tensors of an invariant metric on a Lie group are obtained. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics.

10 
Hypersymplectic structures on Courant algebroids
Volume 7, Number 3, Pages: 255  280, 2015
Paulo Antunes
and Joana M. Nunes da Costa
Abstract
References
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We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a onetoone correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simple way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hypersymplectic structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which are doubles of Lie, quasiLie and protoLie bialgebroids are investigated.

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