ISSN:
 1941-4889

eISSN:
 1941-4897

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Volume 9, 2017

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Volume 7, 2015

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Volume 5, 2013

Volume 4, 2012

Volume 3, 2011

Volume 2, 2010

Volume 1, 2009

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. Infinite-dimensional 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.

Contributions to this journal are published free of charge.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 4 issues a year in March, June, September and December.
  • Publishes online only.
  • Indexed in Science Citation Index-Expanded, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • JGM is a publication of the American Institute of Mathematical Sciences with the support of the Consejo Superior de Investigaciones Científicas (CSIC). All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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Modified equations for variational integrators applied to Lagrangians linear in velocities
Mats Vermeeren
2019, 11(1) : 1-22 doi: 10.3934/jgm.2019001 +[Abstract](154) +[HTML](97) +[PDF](1329.23KB)
Abstract:

Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.

Geometry of Routh reduction
Katarzyna Grabowska and Paweƚ Urbański
2019, 11(1) : 23-44 doi: 10.3934/jgm.2019002 +[Abstract](156) +[HTML](77) +[PDF](504.76KB)
Abstract:

The Routh reduction for Lagrangian systems with cyclic variable is presented as an example of a Lagrangian reduction. It appears that the Routhian, which is a generating object of reduced dynamics, is not a function any more but a section of a bundle of affine values.

Linear phase space deformations with angular momentum symmetry
Claudio Meneses
2019, 11(1) : 45-58 doi: 10.3934/jgm.2019003 +[Abstract](134) +[HTML](58) +[PDF](403.46KB)
Abstract:

Motivated by the work of Leznov-Mostovoy [17], we classify the linear deformations of standard \begin{document}$ 2n $\end{document}-dimensional phase space that preserve the obvious symplectic \begin{document}$ \mathfrak{o}(n) $\end{document}-symmetry. As a consequence, we describe standard phase space, as well as \begin{document}$ T^{*}S^{n} $\end{document} and \begin{document}$ T^{*}\mathbb{H}^{n} $\end{document} with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in \begin{document}$ {\mathbb{R}}^{n+2} $\end{document}.

A geometric perspective on the Piola identity in Riemannian settings
Raz Kupferman and Asaf Shachar
2019, 11(1) : 59-76 doi: 10.3934/jgm.2019004 +[Abstract](141) +[HTML](63) +[PDF](408.79KB)
Abstract:

The Piola identity \begin{document}$ \operatorname{div}\; \operatorname{cof} \;\nabla f = 0 $\end{document} is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.

A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems
Michał Jóźwikowski and Witold Respondek
2019, 11(1) : 77-122 doi: 10.3934/jgm.2019005 +[Abstract](111) +[HTML](74) +[PDF](737.83KB)
Abstract:

We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of generalized variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective. In particular, it allows to derive, interpret and generalize many known results on non-Abelian Chaplygin systems. We apply it also to a class of systems on Lie groups with a left-invariant constraints distribution. Concrete examples of the unicycle in a potential field, the two-wheeled carriage and the generalized Heisenberg system are discussed.

Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics
Manuel de León, Juan Carlos Marrero and David Martín de Diego
2010, 2(2) : 159-198 doi: 10.3934/jgm.2010.2.159 +[Abstract](1466) +[PDF](475.8KB) Cited By(30)
Nonholonomic Hamilton-Jacobi equation and integrability
Tomoki Ohsawa and Anthony M. Bloch
2009, 1(4) : 461-481 doi: 10.3934/jgm.2009.1.461 +[Abstract](1254) +[PDF](789.5KB) Cited By(17)
On Euler's equation and 'EPDiff'
David Mumford and Peter W. Michor
2013, 5(3) : 319-344 doi: 10.3934/jgm.2013.5.319 +[Abstract](1284) +[PDF](661.6KB) Cited By(16)
Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
Ioan Bucataru and Matias F. Dahl
2009, 1(2) : 159-180 doi: 10.3934/jgm.2009.1.159 +[Abstract](1515) +[PDF](318.8KB) Cited By(16)
Integrable Euler top and nonholonomic Chaplygin ball
Andrey Tsiganov
2011, 3(3) : 337-362 doi: 10.3934/jgm.2011.3.337 +[Abstract](1156) +[PDF](488.1KB) Cited By(15)
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
Joachim Escher and Boris Kolev
2014, 6(3) : 335-372 doi: 10.3934/jgm.2014.6.335 +[Abstract](1205) +[PDF](587.5KB) Cited By(15)
Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras
Andrew N. W. Hone and Matteo Petrera
2009, 1(1) : 55-85 doi: 10.3934/jgm.2009.1.55 +[Abstract](1249) +[PDF](494.1KB) Cited By(14)
Clebsch optimal control formulation in mechanics
François Gay-Balmaz and Tudor S. Ratiu
2011, 3(1) : 41-79 doi: 10.3934/jgm.2011.3.41 +[Abstract](1186) +[PDF](597.2KB) Cited By(14)
$G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball
Simon Hochgerner and Luis García-Naranjo
2009, 1(1) : 35-53 doi: 10.3934/jgm.2009.1.35 +[Abstract](1237) +[PDF](272.6KB) Cited By(13)
Geodesic Vlasov equations and their integrable moment closures
Darryl D. Holm and Cesare Tronci
2009, 1(2) : 181-208 doi: 10.3934/jgm.2009.1.181 +[Abstract](1262) +[PDF](360.8KB) Cited By(12)
Lagrange-d'alembert-poincaré equations by several stages
Hernán Cendra and Viviana A. Díaz
2018, 10(1) : 1-41 doi: 10.3934/jgm.2018001 +[Abstract](1210) +[HTML](358) +[PDF](622.49KB) PDF Downloads(125)
Classical field theory on Lie algebroids: Multisymplectic formalism
Eduardo Martínez
2018, 10(1) : 93-138 doi: 10.3934/jgm.2018004 +[Abstract](1259) +[HTML](430) +[PDF](708.21KB) PDF Downloads(119)
The projective Cartan-Klein geometry of the Helmholtz conditions
Carlos Durán and Diego Otero
2018, 10(1) : 69-92 doi: 10.3934/jgm.2018003 +[Abstract](1088) +[HTML](322) +[PDF](437.43KB) PDF Downloads(91)
On some aspects of the discretization of the suslov problem
Fernando Jiménez and Jürgen Scheurle
2018, 10(1) : 43-68 doi: 10.3934/jgm.2018002 +[Abstract](1016) +[HTML](303) +[PDF](1325.47KB) PDF Downloads(80)
Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control
Leonardo Colombo
2017, 9(1) : 1-45 doi: 10.3934/jgm.2017001 +[Abstract](1088) +[HTML](13) +[PDF](680.5KB) PDF Downloads(80)
Generalized variational calculus for continuous and discrete mechanical systems
Viviana Alejandra Díaz and David Martín de Diego
2018, 10(4) : 373-410 doi: 10.3934/jgm.2018014 +[Abstract](448) +[HTML](219) +[PDF](644.95KB) PDF Downloads(67)
A note on time-optimal paths on perturbed spheroid
Piotr Kopacz
2018, 10(2) : 139-172 doi: 10.3934/jgm.2018005 +[Abstract](1051) +[HTML](261) +[PDF](3595.86KB) PDF Downloads(66)
Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum
Richard Cushman and Jędrzej Śniatycki
2018, 10(4) : 419-443 doi: 10.3934/jgm.2018016 +[Abstract](371) +[HTML](225) +[PDF](1324.84KB) PDF Downloads(66)
On some aspects of the geometry of non integrable distributions and applications
Miguel-C. Muñoz-Lecanda
2018, 10(4) : 445-465 doi: 10.3934/jgm.2018017 +[Abstract](393) +[HTML](240) +[PDF](436.46KB) PDF Downloads(64)
The Euler-Poisson equations: An elementary approach to integrability conditions
Sasho Popov and Jean-Marie Strelcyn
2018, 10(3) : 293-329 doi: 10.3934/jgm.2018011 +[Abstract](591) +[HTML](249) +[PDF](564.97KB) PDF Downloads(58)

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