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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, February 2016
1 
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.

2 
A unifying mechanical equation with applications to nonholonomic constraints and dissipative phenomena
Volume 7, Number 4, Pages: 473  482, 2015
E. Minguzzi
Abstract
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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 
Invariant metrics on Lie groups
Volume 7, Number 4, Pages: 517  526, 2015
Gerard Thompson
Abstract
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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.

4 
Geometric arbitrage theory and market dynamics
Volume 7, Number 4, Pages: 431  471, 2015
Simone Farinelli
Abstract
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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.

5 
Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures
Volume 7, Number 4, Pages: 483  515, 2015
Giovanni Rastelli
and Manuele Santoprete
Abstract
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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 
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.

7 
Models for higher algebroids
Volume 7, Number 3, Pages: 317  359, 2015
Michał Jóźwikowski
and Mikołaj Rotkiewicz
Abstract
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Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a differential relation of special kind (a Zakrzewski morphism). In the paper we investigate basic properties of higher algebroids and present some examples.

8 
Lie algebroids generated by cohomology operators
Volume 7, Number 3, Pages: 295  315, 2015
Dennise GarcíaBeltrán,
José A. Vallejo
and Yurii Vorobiev
Abstract
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By studying the FrölicherNijenhuis decomposition of cohomology operators (that is,
derivations $D$ of the exterior algebra $\Omega (M)$ with $\mathbb{Z}$degree $1$ and $D^2=0$),
we describe new examples of Lie algebroid structures on the tangent bundle $TM$
(and its complexification $T^{\mathbb{C}}M$)
constructed from preexisting geometric ones such as foliations,
complex, product or tangent structures.
We also describe a class of Lie algebroids on tangent bundles associated to idempotent endomorphisms with nontrivial Nijenhuis torsion.

9 
Hypersymplectic structures on Courant algebroids
Volume 7, Number 3, Pages: 255  280, 2015
Paulo Antunes
and Joana M. Nunes da Costa
Abstract
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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.

10 
Twocomponent higher order CamassaHolm systems with fractional inertia operator: A geometric approach
Volume 7, Number 3, Pages: 281  293, 2015
Joachim Escher
and Tony Lyons
Abstract
References
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In the following we study the qualitative properties of solutions
to the geodesic flow induced by a higher order twocomponent CamassaHolm system.
In particular, criteria to ensure the existence of temporally global solutions are presented. Moreover
in the metric case, and for inertia operators of order higher than three, the flow is shown
to be geodesically complete.

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